Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms
-
Ariel E. Barton
and
Michael J. Duffy
Abstract
We establish the Caccioppoli inequality, a reverse Hölder inequality in the spirit of the classic estimate of Meyers, and construct the fundamental solution for linear elliptic differential equations of order
1 Introduction
There is at present a very extensive theory for second-order linear elliptic differential operators without lower order terms. Such an operator
where
and operators with lower order terms
where
Operators of higher order (2) with variable coefficients
In this article, we will combine the two approaches and investigate operators
Specifically, three of the foundational results of the theory of elliptic operators of the form (1), which have all received considerable study in the cases of operators of the forms (2) and (3), are Caccioppoli’s inequality, Meyers’s reverse Hölder inequality for gradients, and the fundamental solution. In this article we investigate these three topics in the case of operators of the form (4) under certain assumptions on the coefficients.
For operators (1) or (2) without lower order terms, it is usual to require that all coefficients be bounded. Applying Hölder’s inequality yields the bound
for any
Observe that elements of
The Gagliardo-Nirenberg-Sobolev inequality gives a natural normalization condition on
In [50], the authors introduced the function space
The Gagliardo-Nirenberg-Sobolev inequality gives a natural isomorphism between
We wish to consider higher smoothness spaces. An induction argument shows that, if
We define the
Remark 5
If
We will consider operators that satisfy, for all suitable test functions
and the bound
for a range of
(In Section 4, following [4], we will consider operators satisfying a slightly weaker form (34) of the Gårding inequality (6).)
Note that if
There are many possible conditions that can be imposed on the coefficients
where
Remark 9
Recall from Remark 5 that if
The number
Note that if
We will also consider coefficients satisfying Bochner norm estimates
where
Again, for second-order operators (
Like the condition (8), the condition (10) implies bound (7) for a range of
We note that the conditions (8) and (10) differ from those of [33,81], in which the authors investigate the system (3) or (4) for coefficients
1.1 The Caccioppoli inequality and Meyers’s reverse Hölder inequality
The Caccioppoli inequality (established in the early twentieth century) is valid for all operators
It can be generalized to the case
where
Remark 11
It is common to formulate the Caccioppoli inequality (and Meyers’s reverse Hölder inequality below) for solutions to
Specifically, if
Conversely, if
Remark 12
In the case of equations (
The Caccioppoli inequality has been generalized to operators of the form (2) (higher order equations without lower order terms) in [31] and with some refinements in [4,20]. It has been extended to operators of the form (3) (second-order operators with lower order terms) in [36] (see also [30]). In the case of higher order operators with lower order terms of the form (2), a parabolic Caccioppoli inequality was established in [33] under the assumption that all coefficients (including the lower order coefficients) are bounded; this is different from the assumptions of this article.
In [66], Meyers established a reverse Hölder estimate. Specifically, he established that for equations (
The exponent
Caccioppoli’s inequality is still valid for systems of the form (4), that is, higher order equations with lower order terms. The argument is largely that of [20,31] and is presented in Section 4.
The obvious generalization of Meyers’s reverse Hölder inequality is not valid in the case of operators (even second-order operators) with lower order terms. That is, for any given positive integers
with coefficients satisfying the conditions (6) and (8), and a function
and, indeed, the stronger bound
See Section 6.2.
Weaker generalizations have been investigated in [30] and the argument of Section 6 takes many ideas therefrom. The following theorem is the first main result of this article. It will be proven in Sections 4 (the case
Theorem 14
Let
Then there is a
Let
Let
Then
for all
Here,
If
1.2 The fundamental solution
The fundamental solution
Formally, the fundamental solution can be written as
This is the approach taken in both [20] and the present article for general higher order operators of the form (2) or (4). By assumptions (6) and (7) and the Lax-Milgram lemma,
This approach, with some attention to the details and use of the Caccioppoli and Meyers inequalities, yields the following theorem. This theorem is the second main result of the present article.
Theorem 15
Let L be an operator of order
Then there exists a number
Suppose that
Suppose further that
If
where
Furthermore, we have the symmetry property
Finally, suppose that
The integral converges absolutely for almost every
Then there is a function
for all
Many assumptions on the coefficients other than (8) and (10) are reasonable. We construct the fundamental solution in Section 7. In that section, we will not explicitly use the assumptions (8) and (10); instead we will use their consequences, the Caccioppoli and Meyers inequalities, for the operator
1.2.1 Other approaches
The approach of this article and of [20] uses higher order operators, and in particular the higher order Caccioppoli and Meyers inequalities, to construct the fundamental solution, and as such has only been available since the development of a strong theory of higher order operators. The fundamental solution for second-order operators has been of interest for a long time, and other approaches to its construction have been used.
If
In some cases, solutions to
This approach was used to construct the fundamental solution for operators of the form (1) [47,50] and (3) [36,69,75] in dimension
A different approach involving kernels for the heat semigroup
Considerably more work must be expended to apply the semigroup approach in dimension
This approach does require the De Giorgi-Nash property of elliptic operators, or a condition, such as real coefficients, that implies this property. However, this approach often yields stronger estimates than those of the present paper, and indeed stronger estimates than those true of the fundamental solution for the Laplace operator. See, for example, [37,64,76].
1.3 Outline
The outline of this article is as follows. In Section 2, we will define our terminology. We will give some results concerning function spaces (in particular, Sobolev spaces) in Section 3.
We will prove the Caccioppoli inequality in Section 4. We will prove our generalization of Meyers’s reverse Hölder inequality in Section 6.1 and construct the counterexample of the inequality (13) in Section 6.2.
We will construct the fundamental solution in Section 7.
Some results concerning invertibility of the operator
2 Definitions
2.1 Basic notation
We consider divergence-form elliptic systems of
When
As mentioned in Theorem 14, if
We employ the use of multiindices in
for any multiindex
We will use the Leibniz Rule for multiindices, that is, that for all suitably differentiable functions
2.2 Function spaces
Let
if
If
If
with a suitable modification in the case
We define the inhomogeneous Sobolev norm as follows:
where derivatives are required to exist in the weak sense. We then define the homogeneous Sobolev norm as
Observe that by the Poincaré inequality, if
The Sobolev spaces are then the spaces of equivalence classes of locally integrable functions that have weak derivatives whose Sobolev norm is finite, with the equivalence relation
Recall that for
See, for example, [41, Section 5.6]. Notice that
We will now generalize equation (22). Let
When considering elliptic operators of order
If
We then define
Then
We will generally write bounded linear functionals on
Note that if
2.3 Elliptic operators
Let
We define the differential operator
for all appropriate test functions
Remark 27
If
Thus, in this case, we may write
as a classically defined linear differential operator; this coincides with formula (26) if
We define
Definition 30
We let
and such that if
for all
We consider singleton sets to be intervals, so
Remark 32
If
Remark 33
The condition
As noted in Section 1, if
We will consider coefficients that satisfy the Gårding inequality (6). In [4], Auscher and Qafsaoui consider higher order elliptic systems in divergence form in which ellipticity is in the sense of the following weaker Gårding inequality:
where
Throughout, we will let
A standard argument involving the Lax-Milgram lemma (see Lemma 58) shows that if
Definition 35
If
By compatibly invertible, we mean that
Compatibility is not automatically true; see [13] for an example of operators that are invertible, but not compatibly invertible, in some sense.
We will conclude this section by reminding the reader that our main focus is on coefficients that satisfy bound (8), that is,
or bound (10), that is,
where
Elementary computations involving Hölder’s inequality (Lemma 56) shows that both conditions (8) and (10) imply that
3 The Gagliardo-Nirenberg-Sobolev and Poincaré inequalities and their consequences
In this section, we will collect some results regarding Sobolev functions that will be useful throughout the article. These results are mainly consequences of the Gagliardo-Nirenberg-Sobolev inequality and induction arguments.
We will begin with Section 3.1, in which we will consider the global function spaces
We will often wish to consider the behavior of functions in thin annuli. Thus, in Section 3.3, we will establish results in (possibly thin) annuli rather than cubes. We will sometimes need different forms of estimates and so will also investigate the Poincaré inequality in thin annuli.
Finally, in Section 3.4, we will investigate the behavior of Sobolev functions when multiplied by cutoff functions; since our standard cutoff functions have gradients supported in an annulus, this will build on the results of Section 3.3.
3.1 Global Sobolev spaces
In this section, we will establish some basic properties of the spaces
The global Gagliardo-Nirenberg-Sobolev inequality
is true for functions
Theorem 37
Let
Proof
Uniqueness of
Choosing
We may then compute that
Summing, we see that if
and so
Taking the limit as
We now generalize to higher order.
Corollary 38
Suppose that
In particular,
Proof
Recall the definition (23) of
for some
Applying the definitions (21) and (24) of
We will now establish a bound on the Bochner norm of elements of
Corollary 39
Let
In particular, if
Proof
By Corollary 38, we have that
In particular, for almost every
By definition,
and because this quantity is finite, we must have that
for almost every
Fix some
But
and because both
Thus,
for almost every
This completes the proof.□
3.2 Sobolev functions in cubes
In this section, we will establish analogs to Corollaries 38 and 39 in cubes.
Remark 40
In this section and throughout this article we have chosen to work in cubes rather than in balls. This simplifies certain covering arguments (we never need to use the Vitali covering lemma when working with cubes), but the primary motivation is ease of use with Bochner norms. Recall that the
Lemma 41
Let
Proof
Suppose first that
for any function
Iterating this argument with
We may also control Bochner norms; this is very useful in the case that the coefficients satisfy the condition (10).
Lemma 42
Let
Proof
Let
By applying Lemma 41 in dimension
Integrating in
3.3 Sobolev functions in annuli
We will now establish analogs to Lemmas 41 and 42 in cubical annuli, that is, in domains of the form
The rectangles in the proof of Lemma 43.
Lemma 43
Let
Then there is a constant
If in addition
Proof
Observe that there exists an integer
Let
Furthermore, the rectangles in
Suppose
Because
By Lemma 42 in rectangles,
By the triangle inequality in the sequence space
A similar (and simpler) argument establishes the bound on
Lemma 43 generalizes the Gagliardo-Nirenberg-Sobolev inequality to thin annuli. We remark on the presence of the term
The following lemma allows us to bound a function
Lemma 44
Let
Proof
We restrict to the case
Let
Observe that
Let
If
where
If
Applying our aforementioned formula for
Using the chain rule of single variable calculus and reversing our aforementioned arguments,
We will apply this argument to
We also need to integrate the gradient. Let
Note that
and so
Now,
Thus, the Poincaré inequality holds in an annulus with constant independent of
3.4 Sobolev norms and cutoff functions
A particular application of Lemmas 43 and 44 is the following result concerning smooth cutoff functions.
Lemma 45
Let
Let
If
Proof
We begin by using the definition of the
Observe that
By Lemma 43,
This completes the proof.□
4 The Caccioppoli inequality
The Caccioppoli inequality was established first by Caccioppoli in the early twentieth century and is a foundational result used throughout the theory of second-order divergence form equations. It has been generalized to the case of second-order operators with lower order terms in [36], and of higher order equations (without lower order terms) first in [31], and later with some refinements in [4,20].
We now generalize these results to the case of higher order equations with lower order terms. We will follow [4] and derive a Caccioppoli inequality for equations that satisfy the weak Gårding inequality (34) (and not necessarily the stronger Gårding inequality (6)). We will follow [31] and establish the Caccioppoli inequality for solutions
We begin with the following lemma. This lemma was proven first in [31] for operators of order
Lemma 46
Let
Let
where
Proof
Let
Define
We first consider the left-hand side of formula (47). By the Leibniz rule, and separating out the
Now as in [20], we write
for some functions
It is desirable to have our final term in terms of
Similar measures as taken earlier also give us
Thus, by combining the previous two equations and reintroducing summation, we see that
We write this as I = II + III + IV + V. Observe that by formula (47),
By the condition (34), we have that
Suppose that the condition (10) is true. By Hölder’s inequality and properties of
Recall that
Summing, we see that
By Lemma 43,
So
By applying Young’s inequality, we see that
A similar argument with the roles of
The argument in the case that condition (8) is true is similar.
We thus have that
Subtracting the final term and applying formula (49) yields that
By definition of
By Lemma 45 with
By using the Leibniz rule and arguing as earlier, we see that
By Corollary 38,
Subtracting the second term on the right-hand side and observing that
We wish to improve the Caccioppoli inequality by removing the intermediate derivatives (i.e.,
Theorem 51
Let
whenever
Then
for some constant C depending only on m, the dimension d, and the constant
Furthermore, if
Now if we combine Lemma 46 and Theorem 51, we obtain the desired Caccioppoli inequality in which we bound
Corollary 53
Let L be an operator of order
Let
Then we have that
and for all
where C is a constant depending on the dimension d, the order
5 Invertibility of
L
In this section, we will investigate boundedness and invertibility of the operator
We remark that invertibility requires the Gårding inequality (6), and not only the weaker Gårding inequality (34) of Section 4 and [4]; thus, for the remainder of this article, we will always assume the strong Gårding inequality (6).
We will begin with boundedness of
Lemma 56
Let
If
If
If
Proof
If
Similarly, if
Suppose that
Thus by Hölder’s inequality, for such
which by the condition (8) and the definition (24) of
Summing over
Now suppose that
Observe that
Applying Hölder’s inequality first in
Applying Corollary 39 and summing completes the proof.□
We now establish invertibility of
Theorem 57
[14, Theorem 2.1] Let
for every
Lemma 58
Let
Proof
Let
Notice that by formula (6),
Let
for all
We conclude this section by establishing invertibility of
Lemma 62
(Šneǐberg’s lemma [7, Theorem A.1]) Let
Given
If
Lemma 63
Let
Let
Then
In particular, these conditions are satisfied if
Proof
By assumption or by Lemma 58,
By [80, Section 5.2.5],
A straightforward interpolation argument shows that if
Finally, by Šneǐberg’s lemma,
6
L
p
bounds on solutions and their gradients
In [66], Meyers established a reverse Hölder estimate; in the notation of the present article, he established that if
for all suitable functions
Theorem 14 represents a generalization to the case of operators with lower order terms. It follows immediately from the next theorem and Lemma 63. We remark that the
Theorem 64
Let
Let
Let
Then
Here,
Remark 65
If
In the case of operators without lower order terms (in which case
We will also establish an estimate for functions
Theorem 66
Let
Let
Let
Then
Given operators with lower order terms, Theorem 64 cannot be strengthened, as shown in the following example.
Theorem 67
Let
Let
Then there exists a linear operator L of the form (26) with
If
Constant coefficient operators without lower order terms such as
By an elementary (and very well known) argument using the Fourier transform, the operator
We will prove Theorems 64 and 66 in Section 6.1 and prove Theorem 67 in Section 6.2.
6.1 Proof of Theorems 64 and 66
We begin with the following variant of Lemmas 41, 42, and 43 in the case where the exponents on each side are different.
Lemma 68
Let
Then there is a constant
If in addition
Proof
By Hölder’s inequality, it suffices to establish the listed bounds for the endpoint value
By Lemma 43, and because
Recall that we have taken
(in particular,
Another application of Lemma 43 yields
as desired.□
Now, recall from Lemma 45 that if
Lemma 69
Let
If
Let
Let
Then
Furthermore, if
where
We follow the convention that the zero function is a polynomial of negative degree; thus, if
Proof of Lemma 69
Let
Recall that
Let
Let
We remark on the symmetry of our situation:
By the Leibniz rule,
Recall from Lemma 45 that
By definition of
By the Poincaré inequality, and because
Recall that
Thus,
We will now bound the integrals over
Suppose that the coefficients
and
Because
By Lemma 43 with
By Lemma 68 with
and by Lemma 44,
Observe that
Because
Combining all of the aforementioned estimates and the definitions of
This completes the proof in the case where
If instead
From Lemma 69, we have a bound on
Lemma 70
Let
Suppose in addition that
Then there is a constant C depending only on
If
Proof
Let
By the definition of
Furthermore, if
Thus, because
Since
By Lemma 69,
If
Because
If
In either case,
and the proof is complete.□
We may combine Lemma 70 with the Caccioppoli inequality (Lemma 53) to prove Theorem 64 in the case
Lemma 71
Let
Suppose in addition that
Then there is a positive constant
If
Proof
Let
Let
Because
Remark 72
If
Remark 73
If
for
We have now established that Theorem 66 is valid if
Lemma 74
Let
For each
Suppose that, whenever
for some nonnegative constants
Then for every set of numbers
Proof
If
Let
If
We compute that
So we may apply Hölder’s inequality to see that
By Young’s inequality,
If
Recall that
Let
Recall that
Choose
as desired.□
6.2 A counterexample
In this section, we will prove Theorem 67.
Let
Define
Let
Let
for a positive constant
Furthermore, if
We choose
and so
Recall that
(Many possible families of such constants exist.) Similarly, for any
Choose some such family.
Define the coefficients
If
If
Otherwise, let
If
Furthermore, because
This is the classical definition of
It remains only to establish a lower bound on
If
Thus, recalling that
where
Furthermore, if
Thus,
Again using the fact that
If
as desired. This completes the proof of Theorem 67.
7 The fundamental solution
In this section, we will construct the fundamental solution. We will begin in Section 7.1 with local estimates on functions in
7.1 Preliminaries for operators of high order
Recall from the definition (24) of
If
(We emphasize that if
We now show that this gives a normalization in
Lemma 78
Let
Then there is a unique function
for all
Furthermore, if X,
where
Proof
Fix
If
Let
and by Hölder’s inequality and because
By formula (23),
By Morrey’s inequality (see [41, Section 5.6.2]), we may redefine the weak derivative
Let
Averaging
We will consider the cases
We may write
Recall that
Thus,
By Hölder’s inequality, we have that
By formula (23), we have that
Let
We now take
Let
Combining these pointwise bounds on
Combining this bound with bounds (79) and (80) with
Remark 83
We observe that if
Suppose
Let
By bound (82) and Hölder’s inequality, if
and so
7.2 The fundamental solution for operators of high order
We now define a preliminary version of our fundamental solution for operators of high order. If
Definition 84
Let
Let
where
We define the fundamental solution
Remark 85
If
Remark 86
Since
That is,
In the remainder of this subsection, we will establish some basic properties of the fundamental solution; we will establish further properties in Sections 7.3–7.5. We will begin with a symmetry property for the operators
Theorem 87
Let
For all
For every
Finally, if we let
then
Proof
That
If
In particular, observe that by Lemma 78,
Similarly, we have that if
This establishes formula (89).□
We will conclude this section with a preliminary bound on the derivatives of the function
Theorem 90
Let L and q satisfy the conditions of Definition
84. Let
Suppose that
where
Recall from Definition 35 that
Proof of Theorem 90
By Lemma 78, if
and so by invertibility of
By Lemma 78, if
Integration yields bound (91) in this case (for all
By Remark 83, if
Because
We are left with the case
Finally, suppose that
By compatible invertibility of
If
which is bound (91).
In any case, bound (91) holds.□
7.3 Mixed derivatives of the fundamental solution
Recall that
Theorem 94
Let L be an operator of the form (26) with
Let
holds whenever
Suppose that
Then for every compact set
whenever
In particular, if the Caccioppoli inequality (95) is valid for
If
where
The remainder of this subsection will be devoted to the proof of Theorem 94. We remark that if
Let
Let
By the weak definition of derivative and the symmetry relation (88),
We now investigate
Lemma 101
With the aforementioned construction and under the conditions of Theorem
90, if
Proof
Let
By Theorem 90,
This is true for all test functions
By Theorem 90, we have that the quantities
We will need a bound on
Lemma 103
With the aforementioned construction, if L is of the form (26) and
where
Proof
Let
and compute that
By formula (89),
Thus,
Recall that
Because
and by Lemma 78, since
Thus,
If
This completes the proof.□
We have established that
Lemma 105
Let
Then for all
where
In particular, if
Proof
Applying bounds (104) and (95) and Lemma 41 to
By formula (100), the
Combining the aforementioned bounds completes the proof.□
We now prove Theorem 94. The assumptions of Theorem 94 include the assumptions of Theorem 90 and Lemmas 103 and 105 with
The Lebesgue space
and a sequence of positive numbers
Integrating by parts and applying formula (100), we see that if
By using properties of convolutions, we see that
where
and so
Suppose
Applying formula (88) yields bound (98) in the case
The space
where
and satisfies bound (96).
7.4 Extraneous parameters
The fundamental solution
Lemma 106
Let
Suppose that
Let
Then we have that
for almost every
As noted after Theorem 90, if
Under the conditions of Theorem 94, existence and local integrability of the mixed partial derivative is valid. Furthermore, under these conditions, we may combine formulas (107) and (97) to see that if
Proof of Lemma 106
Fix some such
Let
Because
By formula (89),
By duality, if
By applying the definition of weak derivative, we see that
for any smooth functions with disjoint compact support. By the Lebesgue differentiation theorem, formula (107) is valid for almost every
We now consider the dependency of
However, the range
7.4.1 Odd dimensions
In odd dimensions, we will let our fundamental solution be
Note that for general rough coefficients, it may be that
7.4.2 Even dimensions
The situation in even dimensions is more complicated. In this case, if
The points
Theorem 109
Let
Then there exists a function
Furthermore,
Proof
If
for all
Fix
Thus, it suffices to find a function
whenever
We observe that if
The lemma is thus reduced to a variant of the classical result that a curlfree vector field is the gradient of a function.
For each
By Remark 86, formula (88), and bound (92) applied to
Now, fix some cube
Let
We furthermore compute that
Recall that
for any cube
Thus, by Fubini’s theorem
for almost every
as desired.□
7.5 Derivatives of
(
L
∗
)
−
1
Recall from formula (89) that, if
By the Hahn-Banach theorem, if
and where
Thus,
We would like a similar integral formula for the derivatives of
Theorem 113
Let
Let
for some
If
and the integral converges absolutely for almost every
Proof
By bound (92) and Hölder’s inequality,
Let
By Fubini’s theorem, if
and so
as
for almost every
and so formula (114) is valid for almost every
Remark 117
If
We are left with the case where
Since
By bound (96) and Fubini’s theorem, we may interchange the order of integration. Applying Hölder’s inequality first in
The final term is
If
By Hölder’s inequality,
Thus, recalling that there are
Recall from formula (23) that
which converges provided
7.6 The fundamental solution for operators of arbitrary order
In this section, we show how to use the fundamental solution for operators of high order to construct the fundamental solution for operators of arbitrary order.
We begin by defining a suitable higher order operator associated to each lower order operator and investigate its properties.
Lemma 118
Let
Here,
Then:
If
If L is bounded and invertible
If
If
(120)
Proof
We need only consider the case
It is well known (see, e.g., [80, Section 5.2.3]) that the Laplacian is a bounded and invertible operator
Thus,
If in addition
Furthermore,
There are real nonnegative constants
We may rearrange our order of summation to see that
Furthermore,
If
Finally, recall that
and so
Thus, natural conditions on
We now use
Theorem 122
Let L be an operator of order
Suppose in addition that the Caccioppoli-Meyers inequality for
If L satisfies either bound (8) or bound (10), then this condition is true with
Then there exists some array of functions
Suppose that
Suppose further that
for all
Furthermore, we have the symmetry property
for almost every
Finally, let
for some compactly supported
and the integral converges absolutely for almost every
Proof
If
By assumption and Lemma 58,
If
If
If
In either case, by Theorem 94,
Bounds (124) and (125) follow from Theorem 94 and Lemmas 106 and 118. The symmetry property (126) follows from the symmetry property (88) for
We are left with formula (127). This property follows from Theorem 113 if
and so
for almost every
Observe that
Remark 128
Theorem 122 involves conditions on
We now prove the claim. Suppose that
is valid for all
It is well known (see [77, Chapter VI, Section 3]) that there is a bounded, linear extension operator
Choose some
Let
By the Meyers inequality for
By the Poincaré inequality and because
Finally, by using the estimate
7.7 Uniqueness
We have constructed a fundamental solution; we now show that it is unique.
Theorem 129
Let
Then
Proof
By bound (124), we have that
for all sufficiently nice test functions
Acknowledgement
This article is based on the second author’s Ph.D. thesis, written under the direction of the first author. We would like to thank the University of Arkansas for support during the second author’s time in graduate school. We would also like to thank our anonymous referees for their careful reading of this article and for their many helpful comments, which have greatly improved the exposition of this article.
-
Funding information: The authors were both employed by the University of Arkansas during the writing of this article.
-
Conflict of interest: The authors state no conflict of interest.
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Articles in the same Issue
- Research Articles
- Asymptotic properties of critical points for subcritical Trudinger-Moser functional
- The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation
- On some dense sets in the space of dynamical systems
- Sharp profiles for diffusive logistic equation with spatial heterogeneity
- Generic properties of the Rabinowitz unbounded continuum
- Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis
- Multiple solutions of p-fractional Schrödinger-Choquard-Kirchhoff equations with Hardy-Littlewood-Sobolev critical exponents
- Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals
- The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation
- A priori bounds, existence, and uniqueness of smooth solutions to an anisotropic Lp Minkowski problem for log-concave measure
- Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity
- Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem
- Gagliardo-Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaces
- Ground states of Schrödinger systems with the Chern-Simons gauge fields
- Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries
- A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities
- A modified Picone-type identity and the uniqueness of positive symmetric solutions for a prescribed mean curvature problem
- On a version of hybrid existence result for a system of nonlinear equations
- Special Issue: Geometric PDEs and applications
- Preface for the special issue on “Geometric Partial Differential Equations and Applications”
- Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure
- Total mean curvatures of Riemannian hypersurfaces
- On degenerate case of prescribed curvature measure problems
- A curvature flow to the Lp Minkowski-type problem of q-capacity
- Aleksandrov reflection for extrinsic geometric flows of Euclidean hypersurfaces
- A note on second derivative estimates for Monge-Ampère-type equations
- The Lp chord Minkowski problem
- Widths of balls and free boundary minimal submanifolds
- Smooth approximation of twisted Kähler-Einstein metrics
- The exterior Dirichlet problem for the homogeneous complex k-Hessian equation
- A Carleman inequality on product manifolds and applications to rigidity problems
- Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity
- Pinched hypersurfaces are compact
- The spinorial energy for asymptotically Euclidean Ricci flow
- Geometry of CMC surfaces of finite index
- Capillary Schwarz symmetrization in the half-space
- Regularity of optimal mapping between hypercubes
- Special Issue: In honor of David Jerison
- Preface for the special issue in honor of David Jerison
- Homogenization of oblique boundary value problems
- A proof of a trace formula by Richard Melrose
- Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents
- Regularity properties of monotone measure-preserving maps
- Examples of non-Dini domains with large singular sets
- Sharp inequalities for coherent states and their optimizers
- Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms
- Propagation of symmetries for Ricci shrinkers
- Linear extension operators for Sobolev spaces on radially symmetric binary trees
- The Neumann problem on the domain in 𝕊3 bounded by the Clifford torus
- On an effective equation of the reduced Hartree-Fock theory
- Polynomial sequences in discrete nilpotent groups of step 2
- Integral inequalities with an extended Poisson kernel and the existence of the extremals
- On singular solutions of Lane-Emden equation on the Heisenberg group
