The Caccioppoli ultrafunctions
-
Vieri Benci
,
Luigi Carlo Berselli
Abstract
Ultrafunctions are a particular
class of functions defined on a hyperreal
field
1 Introduction
The Caccioppoli ultrafunctions can be considered as a kind generalized functions. In many circumstances, the notion of real function is not sufficient to the needs of a theory and it is necessary to extend it. Among people working in partial differential equations, the theory of distributions of Schwartz is the most commonly used, but other notions of generalized functions have been introduced by Colombeau [13] and Sato [18, 19]. This paper deals with a new kind of generalized functions, called “ultrafunctions”, which have been introduced recently in [2] and developed in [6, 7, 8, 9, 10]. They provide generalized solutions to certain equations which do not have any solution, not even among the distributions.
Actually, the ultrafunctions are pointwise defined on a subset of
The main novelty of this paper is that we introduce the space of Caccioppoli
ultrafunctions
The main peculiarities of the ultrafunctions in
The generalized derivative is a local operator, namely, if
For all
(1.1)The “generalized” Gauss theorem holds for any measurable set A (see Theorem 4.4)
To any distribution
where
The most relevant point, which is not present in the previous approaches to ultrafunctions, is that we are able the extend the notion of partial derivative so that it is a local operator and it satisfies the usual formula valid when integrating by parts, at the price of a suitable extension of the integral as well. In the proof of this fact, the Caccioppoli sets play a fundamental role.
It is interesting to compare the result about the Caccioppoli ultrafunctions with the well-known Schwartz impossibility theorem:
Theorem (Schwartz impossibility theorem).
There does not exist a differential algebra
The ultrafunctions extend the space of distributions; they do not
violate the Schwartz theorem since the Leibniz rule, in general, does
not hold (see Remark 4.9). Nevertheless, we can prove
the integration by parts rule (1.1) and the Gauss’ divergence
theorem (with the appropriate extension
The theory of ultrafunctions makes deep use of the techniques of NSA presented via the notion of Λ-limit. This presentation has the advantage that a reader, which does not know NSA, is able to follow most of the arguments.
In the last section we present some very simple examples to show that the ultrafunctions can be used to perform a precise mathematical analysis of problems which are not tractable via the distributions.
1.1 Plan of the paper
In Section 2, we present a summary of the theory of Λ-limits
and their role in the development of the ultrafunctions using nonstandard
methods, especially in the context of transferring as much as possible
the language of classical analysis. In Section 3,
we define the notion of ultrafunctions, with emphasis on the pointwise integral.
In Section 4, we define the most relevant
notion, namely the generalized derivative, and its connections with
the pointwise integral, together with comparison with the classical
and distributional derivative. In Section 5, we show
how to construct a space satisfying all the properties of the generalized
derivative and integrals. This section is the most technical and can
be skipped in a first reading. Finally, in Section 6,
we present a general result and two very simple variational problem.
In particular, the second problem is very elementary but without solutions
in the standard
1.2 Notations
Let X be a set and let Ω be a subset of
If
For any
If W is a generic function space, its topological dual will be denoted by
We denote by
2 Λ-theory
In this section we present the basic notions of Non-Archimedean Mathematics and of Nonstandard Analysis, following a method inspired by [3] (see also [2] and [6]).
2.1 Non-Archimedean fields
Here, we recall the basic definitions and facts regarding non-Archimedean
fields. In the following,
Definition 2.1.
Let
ξ is infinitesimal if, for all positive
ξ is finite if there exists
ξ is infinite if, for all
Definition 2.2.
An ordered field
It is easily seen that all infinitesimal are finite, that the inverse of an infinite number is a nonzero infinitesimal number, and that the inverse of a nonzero infinitesimal number is infinite.
Definition 2.3.
A superreal field is an ordered field
It is easy to show, due to the completeness of
Definition 2.4.
We say that two numbers
Clearly, the relation
Theorem 2.5.
If
Given a finite number ξ, we denote its standard part by
Definition 2.6.
Let
and the galaxy of ξ is the set of all numbers that are finitely close to it, i.e.
By definition, it follows that the set of infinitesimal numbers is
2.2 The Λ-limit
In this subsection we introduce a particular non-Archimedean field by means of Λ-theory[1] (for complete proofs and further information the reader is referred to [1, 2, 6]). To recall the basics of Λ-theory we have to recall the notion of superstructure on a set (see also [15]):
Definition 2.7.
Let E be an infinite set. The superstructure on E is the set
where the sets
and, for every
Here
Throughout this paper we let
and we order
where
For every
For every
In particular, we will refer to the elements of
if and only if for all
As usual, if a property
Notice that the Λ-topology satisfies these interesting properties:
Proposition 2.8.
If the net
Proof.
Suppose that the net
We argue indirectly and we assume that
This contradicts the fact that
Proposition 2.9.
Assume that
there exists a sequence
We refer to the sequence
Proof.
It follows easily from the definitions. ∎
Example 2.10.
Let
is uniquely defined and there exists a sequence
Definition 2.11.
The set of the hyperreal numbers
Every net
τ is the coarsest topology which satisfies the first property.
The existence of such an
namely, we shall use the up-arrow “
Given
we set
and
Then the following well-known theorem holds:
Theorem 2.12.
The definitions in (2.2) and (2.3) are well posed and
Remark 2.13.
We observe that the field of hyperreal numbers is defined as a sort
of completion of the real numbers. In fact,
The isomorphism resembles the classical one between the real numbers and the equivalence classes of Cauchy sequences. This method is well known for the construction of real numbers starting from rationals.
2.3 Natural extension of sets and functions
For our purposes it is very important that the notion of Λ-limit can be extended to sets and functions (but also to differential and integral operators) in order to have a much wider set of objects to deal with, to enlarge the notion of variational problem and of variational solution.
So we will define the Λ-limit of any bounded net of mathematical
objects in
We will define
Definition 2.14.
For
A mathematical entity (number, set, function or relation) which is the Λ-limit of a net is called internal.
Definition 2.15.
If for all
The set
Notice that, while the Λ-limit of a constant sequence of
numbers gives this number itself, a constant sequence of sets gives
a larger set, namely
Given any set E, we can associate to it two sets: its natural extension
Clearly,
Remark 2.16.
If
which, by Proposition 2.8, always exists in the Alexandrov
compactification
The above equation suggests the following definition.
Definition 2.17.
If X is a topological space equipped with a Hausdorff
topology, and
if there is a net
and
otherwise.
By the above definition we have that
Definition 2.18.
Let
be a net of functions. We define a function
as follows: for every
where
A function which is a Λ-limit is called internal.
In particular, if, for all
we set
The function
2.4 Hyperfinite sets and hyperfinite sums
Definition 2.19.
An internal set is called hyperfinite if it is the Λ-limit
of a net
For example, if
is hyperfinite. Notice that
so we can say that every set is contained in a hyperfinite set.
It is possible to add the elements of an hyperfinite set of numbers (or vectors) as follows: let
be an hyperfinite set of numbers (or vectors); then the hyperfinite sum of the elements of A is defined in the following way:
In particular, if
we use the notation
3 Ultrafunctions
3.1 Caccioppoli spaces of ultrafunctions
Let Ω be an open bounded set in
Definition 3.1.
A space of ultrafunctions modeled over the space
where
So, given any vector space of functions
Definition 3.2.
Given a space of ultrafunctions
It is possible to prove (see e.g. [2]) that every space
of ultrafunctions has a σ-basis. Clearly, if
Now we will introduce a class of spaces of ultrafunctions suitable for most applications. To do this, we need to recall the notion of Caccioppoli set:
Definition 3.3.
A Caccioppoli set
E is a Borel set such that
The number
is called Caccioppoli perimeter of E. From now on, with some abuse of notation, the above expression will be written as follows:
this expression makes sense since “
If
where η is a fixed infinitesimal and m is the Lebesgue measure.
Clearly,
also, if E is a bounded Caccioppoli set,
Definition 3.4.
A set E is called special Caccioppoli set
if it is open, bounded and
Now we can define a space
Definition 3.5.
A function
where
Notice that
Hence, in particular,
Definition 3.6.
The space
The existence of a Caccioppoli space of ultrafunctions will be proved in Section 5.
Remark 3.7.
Usually in the study of PDEs, the function space where to work depends
on the problem or equation which we want to study. The same fact is
true in the world of ultrafunctions. However, the Caccioppoli space
whenever the vector field F and E are sufficiently smooth. Thus
the space
3.2 The pointwise integral
From now on we will denote by
where
Equality (3.2) suggests the following definition:
Definition 3.8.
For any internal function
In the sequel we will refer to
From Definition 3.8, we have that
and, in particular,
But in general these equalities are not true for
we have that
while
However, for any set
in fact,
Then, if
since
As we will see in the following part of this paper, in many cases,
it is more convenient to work with the pointwise integral
Example 3.9.
If
and similarly
of course, the term
The pointwise integral allows us to define the following scalar product:
From now on, the norm of an ultrafunction will be given by
Notice that
Theorem 3.10.
If
Moreover, we have that
3.3 The δ-bases
Next, we will define the delta ultrafunctions:
Definition 3.11.
Given a point
and
Let us see the main properties of the delta ultrafunctions:
Theorem 3.12.
The delta ultrafunction satisfies the following properties:
For every
For every
Proof.
(1) Let
Let us prove that
So
and hence
(2) We have
(3) We have
The proof is complete. ∎
By the definition of Γ, for all
From this it follows readily the following result.
Proposition 3.13.
The set
Let us examine the main properties of the δ-basis.
Proposition 3.14.
The δ-basis satisfies the following properties:
One has
For all
For all
Proof.
(i) This is an immediate consequence of the definition of δ-basis.
(ii) By Theorem 3.10, it follows that
(iii) This is an immediate consequence of (ii). ∎
3.4 The canonical extension of functions
We have seen that every function
Definition 3.15.
If
if
Since
and
Notice that
defined by
Example 3.16.
If
Definition 3.17.
If a function f is not defined on a set
Example 3.18.
By the definition above, for all
If
Theorem 3.19.
Let
Proof.
Fix
We have that (see Example 3.16)
Since
Then
Corollary 3.20.
If
3.5 Canonical splitting of an ultrafunction
In many applications, it is useful to split an ultrafunction u
into a part
If
and
Definition 3.21.
For every ultrafunction u consider the splitting
where
We will refer to
as to the singular set of the ultrafunction u.
Notice that
Example 3.22.
Take
In this case
We conclude this section with the following trivial propositions which, nevertheless, are very useful in applications:
Proposition 3.23.
Let W be a Banach space such that
is the canonical splitting of
and
Moreover, if
then
Proof.
It is an immediate consequence of Proposition 2.9. ∎
If we use the notation introduced in Definition 2.17, the above proposition can be reformulated as follows:
Proposition 3.24.
If
If
An immediate consequence of Proposition 3.23 is the following:
Corollary 3.25.
If
Proof.
Since
By Proposition 3.23, we have that
with
On the other hand,
4 Differential calculus for ultrafunctions
In this section, we will equip the Caccioppoli space of ultrafunctions
4.1 The generalized derivative
If
as follows:
However, it would be useful to extend the operator
Definition 4.1.
The generalized derivative
is an operator defined on a Caccioppoli ultrafunction space
If
For all
If
where
We remark that, in the framework of the theory of Caccioppoli sets,
the classical formula corresponding to (IV) is the following: for all
The existence of a generalized derivative will be proved in Section 5.
Now let us define some differential operators:
Next let us consider the divergence:
Finally, we can define the Laplace operator:
4.2 The Gauss’ divergence theorem
By Definition 4.1 (IV), for any set
and by Definition 4.1 (III),
If we take a vector field
Now, if
Then (4.1) is a generalization of the Gauss’ theorem
which makes sense for any set
First of all we need to generalize the notion of Caccioppoli perimeter
and it is possible to define an
In particular, if the reduced boundary of E coincides with
Then the following definition is a natural generalization:
Definition 4.2.
If A is a measurable subset of Ω, we set
and for all
Remark 4.3.
Notice that
In fact, the left-hand term has been defined as
while the right-hand term is
in particular, if
Theorem 4.4.
If A is an arbitrary measurable subset of Ω, we have that
where
Proof.
By Definition 4.1 (III),
Then, using the definition of
The proof is complete. ∎
Clearly, if
Example 4.5.
If A is the Koch snowflake, then the usual Gauss’ theorem makes
no sense since
4.3 Ultrafunctions and distributions
One of the most important properties of the ultrafunctions is that they can be seen (in some sense that we will make precise in this subsection) as generalizations of the distributions.
Definition 4.6.
The space of generalized distributions on Ω is defined as follows:
where
The equivalence class of u in
Definition 4.7.
Let
We have the following result.
Theorem 4.8.
There is a linear isomorphism
Proof.
For a proof see e.g. [9]. ∎
From now on we will identify the spaces
Moreover, with some abuse of notation, we will write also that
Remark 4.9.
Since an ultrafunction
Moreover, we recall that, by a well-known theorem of Schwartz, any
tempered distribution can be represented as
5 Construction of the Caccioppoli space of ultrafunctions
In this section we will prove the existence of Caccioppoli spaces of ultrafunctions (see Definition 3.6) by an explicit construction.
5.1 Construction of the space
V
Λ
(
Ω
)
In this subsection we will construct a space of ultrafunctions
Definition 5.1.
Given a family of open sets
for all
for all
(5.1)
We we will refer to the family
Let us verify that
Lemma 5.2.
For any finite family of special Caccioppoli sets
Proof.
For any
We claim that
We set
and we denote by
Now set
Lemma 5.3.
The following properties hold true:
If
where
Proof.
It follows trivially by the construction. ∎
The next lemma is a basic step for the construction of the space
Lemma 5.4.
For any
If
If
For any
For any
Proof.
We set
and we denote by ρ a smooth bell shaped function having support
in
so that
if
if
With this construction, claims (1) and (2) are trivially satisfied.
Now, for any
and
It is easy to check that the functions
We set
and
so we have that, for any
and
Finally, we can define the
Namely, if
with
5.2 The σ-basis
In this subsection, we will introduce a σ-basis in such a way
that
Theorem 5.5.
There exists a σ-basis for
Proof.
First we introduce in
For any
where
we have that
is a δ-basis for
and hence, by Lemma 5.4, it follows that
Next, we want to complete this basis and to get a δ-basis
for
notice that this definition is not in contradiction with (5.3)
since in the latter
For every
in fact,
It is not difficult to realize that
we have that
Now let
Clearly, it is a σ-basis for
Notice that if
By the above theorem, the following corollary follows straightforward.
Corollary 5.6.
The set
If
If, for any internal set A, we define
then we have the following result:
Theorem 5.7.
If
Proof.
We have that
Since
By Theorem 5.5 (3),
Then by (5.7),
By this equation and the fact that
it follows that
Moreover, since
by (5.8) we have
5.3 Construction of the generalized derivative
Next we construct a generalized derivative on
We set
and
will denote the orthogonal complement of
hence, any ultrafunction
Now we are able to define the generalized partial derivative for
Definition 5.8.
We define the generalized partial derivative
as follows:
where
with
Moreover, if
where, for any linear operator L,
Remark 5.9.
Notice that if
In fact, if
Theorem 5.10.
The operator
Proof.
Let us prove property (I). If
Set
If
so, if
and hence, if we set
Since
We prove property (II). If
and hence for all
The conclusion follows from the arbitrariness of v.
Next let us prove property (III). First we prove this property if
Next we will compute
Now, we compute
Moreover, the last term in (5.12), changing the order on which the terms are added, becomes
On the right-hand side we can change the name of the variables Q and R:
In the last step we have used the fact that
Comparing (5.11) and the above equation, we get that
Let us prove property (III) in the general case. We have that
hence
Now, replacing u with v and applying property (5.15)
for
Comparing the above equation with (5.16), we get that
The proof is complete. ∎
Before proving property (IV), we need the following lemma:
Lemma 5.11.
The following identity holds true: for all
Proof.
We can write
where
Then we have that
if and only if
Moreover, we have that
if and only if
Otherwise, we have that
Then, by Theorem 5.7,
Then (5.17) holds true. ∎
Corollary 5.12.
The operator
Proof.
The result follows straightforward from (5.17) just taking
6 Some examples
We present a general minimization result and two very basic examples which can be analyzed in the framework of ultrafunctions. We have chosen these examples for their simplicity and also because we can give explicit solutions.
6.1 A minimization result
In this subsection we will consider a minimization problem. Let Ω
be an open bounded set in
in the set
Clearly, the above assumptions are not sufficient to guarantee the existence of a solution, not even in a Sobolev space. We refer to [20] for a survey of this problem in the framework of Sobolev spaces. On the other hand, we have selected this problem since it can be solved in the framework of the ultrafunctions.
More exactly, this problem becomes:
Problem.
Find an ultrafunction
We have the following result:
Theorem 6.1.
If assumptions (1)–(3) are satisfied, then the functional
Moreover, if
Proof.
The proof is based on a standard approximation by finite-dimensional
spaces. Let us observe that, for each finite-dimensional space
Problem.
Find
The above minimization problem has a solution, being the functional
coercive, due to the hypotheses on
Next, we use the very general properties of Λ-limits, as introduced in Section 2.2. We set
Then, taking a generic
The last statement is trivial. ∎
Clearly, under this generality, the solution u could be very wild; however, we can state a regularization result which allows the comparison with variational and classical solutions.
Theorem 6.2.
Let the assumptions of the above theorem hold. If
where
In this case
with
and
Proof.
Under the above hypotheses the minimization problem has an additional
a priori estimate in
This proves the first statement, since obviously
Next, if
Finally, if the minimizer is a function
6.2 The Poisson problem in
ℝ
2
Now we consider the very classical problem
If
and it can be characterized in several ways.
First of all, it is the only solution in the Schwartz space
where
in the space
Each of these characterizations provides a different method to prove its existence.
The situation is completely different when
however, none of the previous characterization makes sense. In fact,
we cannot use equation (6.2), since
On the contrary, using the theory of ultrafunctions, we can treat equation (6.1) independently of the dimension.
First of all, we recall that in equation (6.1) with
Problem.
Find
where
Clearly, the solutions of the above problem are the minimizers of the Dirichlet integral
in the space
with
Some people might be disappointed that u depends on R and it is not a standard function; if this is the case, it is sufficient to take
and call w
the standard solution of the Poisson problem with
Dirichlet boundary condition at
This fact proves that the use of infinite numbers is an advantage which people should not ignore.
6.3 An explicit example
If the assumptions of Theorem 6.2 do not hold true, the
solution could not be related to any standard object. For example,
if
Let us consider, for
among the functions such that
Formally, the Euler equation, if
We recall that, by standard arguments,
Hence, if
since it turns out that
If
Theorem 6.3.
If
where
Proof.
First, we show that the generalized solution has at most one discontinuity.
In fact, for
and a discontinuity of derivatives at some
The function
In the specific case, we have (see Figure 1)
We now show that this is not possible because the functional takes
a lower value on the solution with only a jump at
we observe that
where
The function
Actually, the solution is the one shown in Figure 2. Next we show
that there exists a unique point ξ such that the minimum is attained.
We write the functional
We observe that, for all
To study the behavior of
hence we have that
Consequently, the function
has a unique negative minimum at the point
The function
The function
From this, we deduce that there exists one and only one point
From the sign of
hence in the case that
then
If
hence
In particular, we have that
In particular,
This follows since replacing
which is negative for all
is a parabola with negative minimum. ∎
Remark 6.4.
It could be interesting to study this problem in dimension greater than one, namely, to minimize
in the set
and in particular to investigate the structure of the singular set
of u, both in the general case and in some particular situations
in which it is possible to find explicit solutions (e.g.,
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Artikel in diesem Heft
- 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
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- 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
Artikel in diesem Heft
- 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
