The classical theory of calculus of variations for generalized functions
-
Alexander Lecke
and
Paolo Giordano
Abstract
We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler–Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi’s theorem on conjugate points and Noether’s theorem. We close with an application to low regularity Riemannian geometry.
1 Introduction and motivations
Singular problems in the calculus of variations have longly been studied both in mathematics and in relevant applications (see, e.g., [18, 7, 43, 25] and references therein). In this paper, we introduce an approach to variational problems involving singularities that allows the extension of the classical theory with very natural statements and proofs. We are interested in extremizing functionals which are either distributional themselves or whose set of extremals includes generalized functions. Clearly, distribution theory, being a linear theory, has certain difficulties when nonlinear problems are in play.
To overcome this type of problems, we are going to use the category of generalized smooth functions, see [13, 12, 14, 15]. This theory seems to be a good candidate, since it is an extension of classical distribution theory, which allows the modeling of nonlinear singular problems, while at the same time sharing many nonlinear properties with ordinary smooth functions like the closure with respect to composition and several non-trivial classical theorems of calculus. One could describe generalized smooth functions as a methodological restoration of Cauchy–Dirac’s original conception of generalized function, see [8, 28, 24]. In essence, the idea of Cauchy and Dirac (but also of Poisson, Kirchhoff, Helmholtz, Kelvin and Heaviside) was to view generalized functions as suitable types of smooth set-theoretical maps obtained from ordinary smooth maps depending on suitable infinitesimal or infinite parameters. For example, the density of a Cauchy–Lorentz distribution with an infinitesimal scale parameter was used by Cauchy to obtain classical properties, which nowadays are attributed to the Dirac delta function, cf. [24].
In the present work, the foundation of the calculus of variations is set for functionals defined by arbitrary generalized functions. This in particular applies to any Schwartz distribution and any Colombeau generalized function (see, e.g., [5, 6]), and hence justifies the title of the present paper.
For example, during the last years, the study of low regular Riemannian and Lorentzian geometry was intensified and made a huge amount of progress (cf. [26, 27, 41, 33, 30, 40]).
It was shown that the exponential map is a bi-Lipschitz homeomorphism when metrics
In order to present a possible way out of the aforementioned problems, the singular metric g is embedded as a generalized smooth function.
In this way, the embedding
The structure of the present paper is as follows. We start with an introduction into the setting of generalized smooth functions and give basic notions concerning generalized smooth functions and their calculus that are needed for the calculus of variations (Section 2). The paper is self-contained in the sense that it contains all the statements required for the proofs of calculus of variations we are going to present. If proofs of preliminaries are omitted, we clearly give references to where they can be found. Therefore, to understand this paper, only a basic knowledge of distribution theory is needed.
In Section 3, we obtain some preliminary lemmas regarding the calculus of variations with generalized smooth functions.
The first variation and the notion of critical point will be defined and studied in Section 4.
We prove the fundamental lemma of calculus of variations and the full connection between critical points of a given functional and solutions of the corresponding Euler–Lagrange equation.
In Section 5, we study the second variation and define the notion of local minimizer.
We also extend to generalized functions classical necessary and sufficient conditions to have a minimizer, and we give a proof of the Legendre condition.
In Section 6, we introduce the notion of Jacobi field and extend to generalized functions the definition of conjugate points, so as to prove the corresponding Jacobi theorem.
In Section 7, we extend the classical Noether’s theorem.
We close with an application to
Note that Konjik, Kunzinger and Oberguggenberger [25] already established the calculus of variations in the setting of Colombeau generalized functions, by using a comparable methodological approach. Indeed, generalized smooth functions are related to Colombeau generalized functions, and one could say that the former is a minimal extension of the latter so as to get more general domains for generalized functions, and hence the closure with respect to composition and a better behavior on unbounded sets. However, there are some conceptual advantages in our approach.
(i) Whereas generalized smooth functions are closed with respect to composition, Colombeau generalized functions are not. This forced the authors of [25] to consider only functionals defined using compactly supported Colombeau generalized functions, i.e., functions assuming only finite values, or tempered generalized functions.
(ii) The authors of [25] were forced to consider the so-called compactly supported points
(iii) The theory of generalized smooth functions was developed to be very user friendly, in the sense that one can avoid cumbersome “ε-wise” proofs quite often, whereas the proofs in [25] frequently use this technique. Thus, one could say that some of the proofs based on generalized smooth functions are more “intrinsic” and close to the classical proofs in a standard smooth setting. This allows a smoother approach to this new framework.
(iv) The setting of generalized smooth functions depends on a fixed infinitesimal
net
(v) In [25], only the notion of global minimizer is defined, whereas we define the notion of local minimizer, as in [10], using a natural topology in space of generalized smooth curves.
(vi) We obtain more classical results like the Legendre condition, and the classical results about Jacobi fields and conjugate points.
(vii) The Colombeau generalized functions can be embedded into generalized smooth functions. Thus, our approach is a natural extension of [25].
2 Basic notions
The new ring of scalars
In this work, I denotes the interval
We start by defining the new simple non-Archimedean ring of scalars that extends the real field
Definition 2.1.
Let
If
We say that a net
Let
which we call Robinson–Colombeau ring of generalized numbers, see [38, 5, 6]. We denote the equivalence class
In the following, ρ will always denote a net as in Definition 2.1.
The infinitesimal ρ can be chosen depending on the class of differential equations we need to solve for the generalized functions we are going to introduce, see [16].
For motivations concerning the naturality of
We can also define an order relation on
Even in the case where the order
Topologies on
ℝ
~
n
ρ
On the
Definition 2.2.
Let
We write
For each
The relation
The following result is useful to deal with positive and invertible generalized numbers (cf. [35]).
Lemma 2.3.
Let
x is invertible and
For each representative
For each representative
We will also need the following result.
Lemma 2.4.
Let
Internal and strongly internal sets
A natural way to obtain sharply open, closed and bounded sets in
Definition 2.5.
Let
Let
We say that the internal set
Therefore,
The following theorem shows that internal and strongly internal sets have dual topological properties:
Theorem 2.6.
For
Generalized smooth functions and their calculus
Using the ring
Definition 2.7.
Let
Let us note explicitly that this definition states minimal logical conditions to obtain a set-theoretical map from X into Y, defined by a net of smooth functions.
In particular, the following theorem states that the equality
Theorem 2.8.
Let
For all
Each
Assume that the GSF f is locally Lipschitz in the Fermat topology and that its Lipschitz constants are always finite, i.e.,
Subsets
The differential calculus for GSF can be introduced by showing existence and uniqueness of another GSF serving as incremental ratio.
Theorem 2.9 (Fermat–Reyes theorem for GSF).
Let
There exists a sharp neighborhood T of
Any two generalized incremental ratios coincide on a sharp neighborhood of
We have
If U is a large open set, then an analogous statement holds by replacing sharp neighborhoods by large neighborhoods.
Note that this result permits the consideration of the partial derivative of f with respect to an arbitrary generalized vector
The set of all the j-multilinear maps
The following result follows from the analogous properties for the nets of smooth functions defining f and g.
Theorem 2.10.
Let
For each
Let
We also have a generalization of the Taylor formula.
Theorem 2.11.
Let
If we further assume that all the n components
Formula (2.1) corresponds to a direct generalization of Taylor formulas for ordinary smooth functions with Lagrange remainder.
On the other hand, in (2.2) and (2.3), the possibility that the differential
The following local inverse function theorem will be used in the proof of Jacobi’s theorem (see [13] for a proof).
Theorem 2.12.
Let
We can define right and left derivatives as, e.g.,
Theorem 2.13.
Let
Definition 2.14.
Under the assumptions of Theorem 2.13,
we denote by
All the classical rules of integral calculus hold in this setting:
Theorem 2.15.
Let
Theorem 2.16.
Let
Embedding of Schwartz distributions and Colombeau functions
We finally recall two results that give a certain flexibility in constructing embeddings of Schwartz distributions.
Note that both the infinitesimal ρ and the embedding of Schwartz distributions have to be chosen depending on the problem we aim to solve.
A trivial example in this direction is the ODE
If
Lemma 2.17.
Let
If
It is worth noting that the condition (iv) of null moments is well known in the study of convergence of numerical solutions of singular differential equations, see, e.g., [42, 9, 21] and references therein.
Concerning the embeddings of Schwartz distributions, we have the following result, where
is called the set of compactly supported points in
Theorem 2.18.
Under the assumptions of Lemma 2.17, let
uniquely extends to a sheaf morphism of real vector spaces
and satisfies the following properties:
If
If
Concerning the embedding of Colombeau generalized functions, we recall that the special Colombeau algebra on Ω is defined as the quotient
and the latter is
Using
Theorem 2.19.
A Colombeau generalized function
2.1 Extreme value theorem and functionally compact sets
For GSF, suitable generalizations of many classical theorems of differential and integral calculus hold such as the intermediate value theorem, mean value theorems, a sheaf property for the Fermat topology, local and global inverse function theorems, the Banach fixed point theorem and a corresponding Picard–Lindelöf theorem, see [15, 14, 31, 13].
Even though the intervals
Theorem 2.20.
Let
We shall use the assumptions on K and
Definition 2.21.
A subset K of
If, in addition,
We motivate the name functionally compact subset by noting that on this type of subsets, GSF have properties very similar to those that ordinary smooth functions have on standard compact sets.
Remark 2.22.
By [36, Proposition 2.3], any internal set
If
For the empty set, we have
In the present paper, we need the following properties of functionally compact sets.
Theorem 2.23.
Let
As a corollary of this theorem and Remark 2.22 (ii), we get the following.
Corollary 2.24.
If
Let us note that
Theorem 2.25.
Let
A theory of compactly supported GSF has been developed in [13], and it closely resembles the classical theory of LF-spaces of compactly supported smooth functions. It establishes that for suitable functionally compact subsets, the corresponding space of compactly supported GSF contains extensions of all Colombeau generalized functions, and hence also of all Schwartz distributions.
3 Preliminary results for calculus of variations with GSF
In this section, we study extremal values of generalized functions at sharply interior points of intervals
The following lemma shows that we can interchange integration and differentiation while working with generalized functions.
Lemma 3.1.
Let
Proof.
We first note that
This proves that also the left-hand side of (3.1) is well defined as a derivative of a GSF. From the classical derivation under the integral sign, the Fermat–Reyes Theorem 2.9, and Theorem 2.13 about definite integrals of GSF, we obtain
The next result will be used frequently.
Lemma 3.2.
Let
Proof.
Note that the internal set
Remark 3.3.
If
Assume that x,
Then taking
Definition 3.4.
We call
Lemma 3.5.
Let
Proof.
By Lemma 2.3, it follows that for
Analogously to the classical case, we say that
Lemma 3.6.
Let
Proof.
Without loss of generality, we can assume
Set
Thus,
As a consequence of Lemma 2.4 and Theorem 2.8 (iv), we have the following lemma.
Lemma 3.7.
Let
Now, we are able to prove the “second-derivative-test” for GSF.
Lemma 3.8.
Let
Proof.
As above, we can assume that
By assumption, for all
By Lemma 3.6, we know that
Therefore, also
Now assume that
where
Thus,
We can hence write
For the generalization of Lemmas 3.6 and 3.8 to the multivariate case, one can proceed as above, using the ideas of [25]. Note, however, that we do not need this generalization in the present work.
4 First variation and critical points
In this section, we define the first variation of a functional and prove that some classical results have their counterparts in this generalized setting, for example, the fundamental lemma (Lemma 4.4) or the connection between critical points and the Euler–Lagrange equations (Theorem 4.5).
Definition 4.1.
If
When the use of the points
One of the positive features of the use of GSF for the calculus of variations is their closure with respect to composition. For this reason, the next definition of functional is formally equal to the classical one, though it can be applied to arbitrary generalized functions F and u.
Theorem 4.2.
Let
Let also
Proof.
Using Theorems 2.10 and 2.15, and Lemma 3.1, we have
We call
To prove the fundamental lemma of the calculus of variations, Lemma 4.4, we first show that every GSF can be approximated using generalized strict delta nets.
Lemma 4.3.
Let
For t small,
Then
Proof.
We only have to generalize the classical proof concerning limits of convolutions with strict delta nets. We first note that
and so these integrals exist because
For each
The right-hand side of (4.2) can be taken arbitrarily
small in
Lemma 4.4 (Fundamental lemma of the calculus of variations).
Let
then
Proof.
Let
and hence Lemma 4.3 yields
Thus, we obtain the following theorem.
Theorem 4.5.
Let
for I given by (4.1), if and only if
5 Second variation and minimizers
As in the classical case (see, e.g., [10]), thanks to the extreme value Theorem 2.20 and the property of the interval
Definition 5.1.
Let
where
The following result permits the calculation of the (generalized) norm
Lemma 5.2.
Under the assumptions of Definition 5.1, let
If the net
For all
Proof.
By the standard extreme value theorem applied ε-wise, we get
the existence of
Hence,
Thus,
But
This proves both that
Using these
Definition 5.3.
Let
We set
If
As in [15, Theorem 2], one can easily prove that sharply open sets form a topology on
Definition 5.4.
Let
For all
Note that
We say that u is a local minimizer of I in
(5.1)We define the second variation of I in the direction
Note also explicitly that the points p,
which we abbreviate as
The following results establish classical necessary and sufficient conditions to decide if a function u is a minimizer for the given functional (4.1).
Theorem 5.5.
Let
Proof.
Let
Theorem 5.6.
Let
Then u is a local minimizer of the functional I in
Moreover, if
Proof.
For any
for some
which is our conclusion.
Note explicitly that if
Now, assume that
Lemma 5.7.
Let
Proof.
We can apply the integral mean value theorem for each ε and each defining net
Now, we take the limit for
We now derive the so-called necessary Legendre condition.
Theorem 5.8.
Let
![Figure 1
This figure illustrates the function
ϑ
ε
{\vartheta_{\varepsilon}}
we are considering (blue). The dotted green triangle symbolizes the function which is used in the classical proofs of the Legendre necessary condition (cf. [23, Theorem 1.3.2]).](/document/doi/10.1515/anona-2017-0150/asset/graphic/j_anona-2017-0150_fig_0001.png)
This figure illustrates the function
Proof.
Let
The net
Now, setting
where we used the evident notation
Note that there always exists
Using Lemmas 5.7 and 3.2, (5.4) and (5.3), we obtain
But (iii) yields
6 Jacobi fields
As in the classical case, Theorem 5.5 (ii) motivates us to define the accessory integral
where
for all
As usual, we note that
In other words,
Since u is given, (6.2) is an
Definition 6.1.
A solution
The following result confirms that the intuitive interpretation of a Jacobi field as the tangent space of a smooth family of solutions of the Euler–Lagrange equation still holds in this generalized setting.
Lemma 6.2.
Let
Then
is a Jacobi field along u.
Proof.
A straight forward calculation gives
6.1 Conjugate points and Jacobi’s theorem
The classical key result concerning Jacobi fields relates conjugate points and minimizers. The main aim of the present section is to derive this theorem in our generalized framework, by extending the ideas of the proof of [23, Theorem 1.3.4].
A crucial notion is hence that of piecewise GSF.
Definition 6.3.
We call piecewise GSF an n-tuple
For all
For all
Every pointwise GSF
For all
We also use the arrow notation
Remark 6.4.
Since the order relation
If
Every GSF
If
If
Piecewise GSF inherit from their defining components a well-behaved differential and integral calculus. The former is even more general and taken from [1].
Definition 6.5.
Let
It is worth noting that
in the sharp topology. The following definition is based on [1, Definition 2.2].
Definition 6.6.
Let
In this case, using Landau little-oh notation, we can hence write
As in the classical case, (6.3) implies the uniqueness of
In particular, this notion of derivative applies to the set-theoretical function induced by a piecewise GSF
The notion of definite integral is naturally introduced in the following definition.
Definition 6.7.
Let
Since our main aim in using piecewise GSF is to prove Jacobi’s theorem, we do not need to prove that the usual elementary rules of integration hold, since we will always reduce to integrals of GSF.
Having a notion of derivative and of definite integral also for piecewise GSF allows to study functionals of the form
This leads to the following natural definition: we say that a piecewise GSF v is a piecewise GSF (global) minimizer if
Classically, several proofs of Jacobi’s theorem use both some form of implicit function theorem and of uniqueness of solution for linear ODE.
Theorem 6.8 (Implicit function theorem).
Let
Moreover, the function
Proof.
The usual deduction of the implicit function theorem from the inverse function theorem in Banach spaces can be easily adapted by using Theorem 2.12 and noting that
In the next theorem, the dependence of the entire theory on the initial infinitesimal net
Theorem 6.9 (Solution of first order linear ODE).
Let
where
Moreover, this y is given by
Proof.
We first note that
where
Therefore, all values of
To show uniqueness, we can proceed as in the smooth case.
Assume that
From the uniqueness of primitives of GSF and Theorem 2.13, we have that
If α,
The following result is the key regularity property that is needed to prove Jacobi’s theorem.
Lemma 6.10.
Let
Finally, assume that
In particular, if
Proof.
Set
Our assumption on the invertibility of
But we have
Therefore, the uniqueness in (6.8) yields
We now integrate the Euler–Lagrange equation (6.7)
on
This entails that we can write
But the function
and this goes to 0 as
Applying this equality in (6.9), we get
In the following definition and below, we use the complete notation
Definition 6.11.
Let
Jacobi’s theorem shows that we cannot have minimizers if there are interior points conjugate to a. In order to prove it in the present generalized context, we finally need the following lemma.
Lemma 6.12.
Let
Proof.
Since ψ is
Thus, by integration by parts, we calculate
where we used the fact that η is a Jacobi field. ∎
After these preparations, we can finally prove Jacobi’s theorem.
Theorem 6.13 (Jacobi).
Let
For all
Then u cannot be a local minimizer of I.
Therefore, for any
Proof.
By contradiction, assume that u is a local minimizer, and let
Thus, Theorem 5.5 (necessary condition for u being a minimizer) gives
This shows that
By the fundamental Lemma 4.4, this implies that η satisfies the Euler–Lagrange equations for ψ in the interval
For all
so that we can re-write the Jacobi equations (6.2) for η on
By assumptions (iii), we obtain
Note that if one of the quantities in (iii) depends even only polynomially on ε, then we are forced to take, e.g.,
7 Noether’s theorem
In this section, we state and prove Noether’s theorem by following the lines of [3].
We first note that any
For all
In our proofs, we will in fact only use properties (i) and (iii).
The proof of Noether’s theorem is classically anticipated by the following time-independent version, which the general case is subsequently reduced to.
Theorem 7.1.
Let
Suppose that 0 is a sharply interior point of
K is invariant under
(7.2)
Then, the quantity
is constant in
Proof.
We first note that both sides of (7.2) are in
Since the Euler–Lagrange equations (7.1) for K are given by
which is our conclusion by the uniqueness part of Theorem 2.13. ∎
We are now able to prove Noether’s theorem. For the convenience of the reader, in its statement and proof we use the variables t, T, l, L, v, V so as to recall tempus, locus, velocitas, respectively.
Theorem 7.2 (Noether).
Let
Then, the quantity
is constant in
Proof.
Since (7.3) is a GSF in
and
and
From these, for all
Therefore, since u satisfies the Euler–Lagrange equations for F, this entails that w is a solution of the analogous equations for K in
Moreover, (ii) gives
We can hence apply Theorem 7.1, and from (7.5) and (7.6), we get that
is constant in
8 Application to
𝒞
1
,
1
Riemannian metric
In the following, we apply what we did so far to the problem of length minimizers in
In this section, we fix an embedding
By [27, Remark 2.6.2], it follows that we can always find a net of smooth functions
Definition 8.1.
We say that
This definition includes also the possibility that the point p or the vector v could be infinite.
By Theorem 2.19, it follows that if we consider only finite p and v, then any geodesic
The definition of length of a (non-singular) curve needs the following.
Remark 8.2.
We set
Lemma 2.3 readily implies that
Definition 8.3.
Let
Moreover, for
Let
Note that (8.1) are the usual geodesic equations for the generalized metric
We are interested only in global minimizers of the functional
Lemma 8.4.
Let
such that
Proof.
We calculate
By assumption,
We hence obtain the claim by the triangle inequality and the convergence of
Now, we consider
be a solution of the geodesic equation
Let
Using these initial conditions, for each fixed ε, we can solve the following problem:
for a unique
Lemma 8.5.
Let u and
For ε sufficiently small, the solution
The net
Proof.
(i)–(ii) For all i, j, we have that
(iii) If
so that there exists a polynomial P such that
If
Finally, the standard part of the generalized length of y is the length of u.
Theorem 8.6.
Let u and
Proposition 8.7.
Let
Proof.
Let
Corollary 8.8.
Let
Corollary 8.8 gives us a way to answer the question if a certain classical geodesic between two given classical points p and q is a length-minimizer.
Furthermore, we are able to prove the following theorem, relating generalized minimizers to classical minimizers.
Theorem 8.9.
Let
Proof.
Assume to the contrary that there exists a curve
But, by assumption, we have that
This is a contradiction. ∎
9 Conclusions
We can summarize the present work as follows.
(i) The setting of GSF allows to treat Schwartz distributions more closely to classical smooth functions. The framework is so flexible and the extensions of classical results are so natural in many ways that one may treat it like smooth functions.
(ii) One key step of the theory is the change of the ring of scalars into a non-Archimedean one, and the use of the strict order relation
(iii) Others crucial properties are the closure of GSF with respect to composition and the use of the gauge ρ, because they do not force to narrow the theory into particular cases.
(iv) The present extension of the classical theory of calculus of variations
shows that the use of GSF is a powerful analytical technique.
The final application shows how to use them as a method to address problems in an Archimedean setting based on the real field
Concerning possible future developments, we note the following.
(v) A generalization of the whole construction to piecewise GSF seems possible.
(vi) A more elegant approach for the integration of piecewise GSF could use the existence of right and left limits of
extended to
The present work could lay the foundations for further works concerning the possibility to extend other results of the calculus of variations in this generalized setting.
Funding source: Austrian Science Fund
Award Identifier / Grant number: M1876-N35
Award Identifier / Grant number: P25311-N25
Award Identifier / Grant number: P25116-N25
Funding statement: Alexander Lecke has been supported by the uni:doc fellowship programme of the University of Vienna. Lorenzo Luperi Baglini has been supported by grant M1876-N35 of the Austrian Science Fund FWF. Paolo Giordano has been supported by grants P25311-N25 and P25116-N25 of the Austrian Science Fund FWF.
Acknowledgements
We would like to thank M. Kunzinger for helpful discussions and several suggestions that have led to considerable improvements.
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Articles in the same Issue
- Frontmatter
- Asymptotic behavior of evolution systems in arbitrary Banach spaces using general almost periodic splittings
- Solvability of a product-type system of difference equations with six parameters
- On Dirichlet problem for fractional p-Laplacian with singular non-linearity
- Absence of Lavrentiev gap for non-autonomous functionals with (p,q)-growth
- On a class of fully nonlinear parabolic equations
- On sign-changing solutions for (p,q)-Laplace equations with two parameters
- Weighted Caffarelli–Kohn–Nirenberg type inequalities related to Grushin type operators
- On the fractional p-Laplacian equations with weight and general datum
- An elliptic equation with an indefinite sublinear boundary condition
- Liouville-type theorems for elliptic equations in half-space with mixed boundary value conditions
- Well/ill-posedness for the dissipative Navier–Stokes system in generalized Carleson measure spaces
- Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems
- Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach
- A multiplicity result for asymptotically linear Kirchhoff equations
- Higher-order anisotropic models in phase separation
- Well-posedness and maximum principles for lattice reaction-diffusion equations
- Existence of a bound state solution for quasilinear Schrödinger equations
- Existence and concentration behavior of solutions for a class of quasilinear elliptic equations with critical growth
- Homoclinics for strongly indefinite almost periodic second order Hamiltonian systems
- A new method for converting boundary value problems for impulsive fractional differential equations to integral equations and its applications
- Diffusive logistic equations with harvesting and heterogeneity under strong growth rate
- On viscosity and weak solutions for non-homogeneous p-Laplace equations
- Periodic impulsive fractional differential equations
- A result of uniqueness of solutions of the Shigesada–Kawasaki–Teramoto equations
-
Solutions of vectorial Hamilton–Jacobi equations are rank-one absolute minimisers in
- Large solutions to non-divergence structure semilinear elliptic equations with inhomogeneous term
- The elliptic sinh-Gordon equation in a semi-strip
- The Gelfand problem for the 1-homogeneous p-Laplacian
- Boundary layers to a singularly perturbed Klein–Gordon–Maxwell–Proca system on a compact Riemannian manifold with boundary
- Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth
- Multiple solutions for an elliptic system with indefinite Robin boundary conditions
- New solutions for critical Neumann problems in ℝ2
- A fractional Kirchhoff problem involving a singular term and a critical nonlinearity
- Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system
- Characterizing the strange term in critical size homogenization: Quasilinear equations with a general microscopic boundary condition
- Nonlocal perturbations of the fractional Choquard equation
- A pathological example in nonlinear spectral theory
- Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four
- On Cauchy–Liouville-type theorems
- Maximal Lp -Lq regularity to the Stokes problem with Navier boundary conditions
- Besov regularity for solutions of p-harmonic equations
- The classical theory of calculus of variations for generalized functions
- On the Cauchy problem of a degenerate parabolic-hyperbolic PDE with Lévy noise
- Hölder gradient estimates for a class of singular or degenerate parabolic equations
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Critical and subcritical fractional Trudinger–Moser-type inequalities on
- Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential
- Quantization of energy and weakly turbulent profiles of solutions to some damped second-order evolution equations
- An elliptic system with logarithmic nonlinearity
- The Caccioppoli ultrafunctions
- Equilibrium of a production economy with non-compact attainable allocations set
- Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity
- The higher integrability of weak solutions of porous medium systems
- Classification of stable solutions for boundary value problems with nonlinear boundary conditions on Riemannian manifolds with nonnegative Ricci curvature
- Regularity results for p-Laplacians in pre-fractal domains
- Carleman estimates and null controllability of a class of singular parabolic equations
- Limit profiles and uniqueness of ground states to the nonlinear Choquard equations
- On a measure of noncompactness in the space of regulated functions and its applications
- p-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities
- On the well-posedness of a multiscale mathematical model for Lithium-ion batteries
- Global existence of a radiative Euler system coupled to an electromagnetic field
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On the existence of a weak solution for some singular
- Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth
- Clustered solutions for supercritical elliptic equations on Riemannian manifolds
- Ground state solutions for the Hénon prescribed mean curvature equation
- Quasilinear equations with indefinite nonlinearity
- Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition
- Retraction of: Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition
Articles in the same Issue
- Frontmatter
- Asymptotic behavior of evolution systems in arbitrary Banach spaces using general almost periodic splittings
- Solvability of a product-type system of difference equations with six parameters
- On Dirichlet problem for fractional p-Laplacian with singular non-linearity
- Absence of Lavrentiev gap for non-autonomous functionals with (p,q)-growth
- On a class of fully nonlinear parabolic equations
- On sign-changing solutions for (p,q)-Laplace equations with two parameters
- Weighted Caffarelli–Kohn–Nirenberg type inequalities related to Grushin type operators
- On the fractional p-Laplacian equations with weight and general datum
- An elliptic equation with an indefinite sublinear boundary condition
- Liouville-type theorems for elliptic equations in half-space with mixed boundary value conditions
- Well/ill-posedness for the dissipative Navier–Stokes system in generalized Carleson measure spaces
- Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems
- Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach
- A multiplicity result for asymptotically linear Kirchhoff equations
- Higher-order anisotropic models in phase separation
- Well-posedness and maximum principles for lattice reaction-diffusion equations
- Existence of a bound state solution for quasilinear Schrödinger equations
- Existence and concentration behavior of solutions for a class of quasilinear elliptic equations with critical growth
- Homoclinics for strongly indefinite almost periodic second order Hamiltonian systems
- A new method for converting boundary value problems for impulsive fractional differential equations to integral equations and its applications
- Diffusive logistic equations with harvesting and heterogeneity under strong growth rate
- On viscosity and weak solutions for non-homogeneous p-Laplace equations
- Periodic impulsive fractional differential equations
- A result of uniqueness of solutions of the Shigesada–Kawasaki–Teramoto equations
-
Solutions of vectorial Hamilton–Jacobi equations are rank-one absolute minimisers in
- Large solutions to non-divergence structure semilinear elliptic equations with inhomogeneous term
- The elliptic sinh-Gordon equation in a semi-strip
- The Gelfand problem for the 1-homogeneous p-Laplacian
- Boundary layers to a singularly perturbed Klein–Gordon–Maxwell–Proca system on a compact Riemannian manifold with boundary
- Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth
- Multiple solutions for an elliptic system with indefinite Robin boundary conditions
- New solutions for critical Neumann problems in ℝ2
- A fractional Kirchhoff problem involving a singular term and a critical nonlinearity
- Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system
- Characterizing the strange term in critical size homogenization: Quasilinear equations with a general microscopic boundary condition
- Nonlocal perturbations of the fractional Choquard equation
- A pathological example in nonlinear spectral theory
- Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four
- On Cauchy–Liouville-type theorems
- Maximal Lp -Lq regularity to the Stokes problem with Navier boundary conditions
- Besov regularity for solutions of p-harmonic equations
- The classical theory of calculus of variations for generalized functions
- On the Cauchy problem of a degenerate parabolic-hyperbolic PDE with Lévy noise
- Hölder gradient estimates for a class of singular or degenerate parabolic equations
-
Critical and subcritical fractional Trudinger–Moser-type inequalities on
- Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential
- Quantization of energy and weakly turbulent profiles of solutions to some damped second-order evolution equations
- An elliptic system with logarithmic nonlinearity
- The Caccioppoli ultrafunctions
- Equilibrium of a production economy with non-compact attainable allocations set
- Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity
- The higher integrability of weak solutions of porous medium systems
- Classification of stable solutions for boundary value problems with nonlinear boundary conditions on Riemannian manifolds with nonnegative Ricci curvature
- Regularity results for p-Laplacians in pre-fractal domains
- Carleman estimates and null controllability of a class of singular parabolic equations
- Limit profiles and uniqueness of ground states to the nonlinear Choquard equations
- On a measure of noncompactness in the space of regulated functions and its applications
- p-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities
- On the well-posedness of a multiscale mathematical model for Lithium-ion batteries
- Global existence of a radiative Euler system coupled to an electromagnetic field
-
On the existence of a weak solution for some singular
- Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth
- Clustered solutions for supercritical elliptic equations on Riemannian manifolds
- Ground state solutions for the Hénon prescribed mean curvature equation
- Quasilinear equations with indefinite nonlinearity
- Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition
- Retraction of: Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition
