Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space
-
Andrea Cianchi
,
Gael Y. Diebou
Abstract
Existence and global regularity results for boundary-value problems of Robin type for harmonic and polyharmonic functions in n-dimensional half-spaces are offered.
The Robin condition on the normal derivative on the boundary of the half-space is prescribed by a nonlinear function
1 Introduction
We deal with the existence and global regularity of harmonic and polyharmonic functions in n-dimensional half-spaces, subject to general nonlinear Robin-type boundary conditions.
The problems under consideration for harmonic functions have the form
Here
If
has also been investigated for subcritical (
The punctum of the present contribution is that the function
for some
The problem (1.1), with
for some
This functional is well defined for
The approach to the problem (1.1) adopted in this paper has a non-variational nature and is based on a combination of potential theoretic techniques, sharp non-standard results from the theory of Sobolev-type spaces, and fixed-point arguments. Novel developments on these methods in the framework of Orlicz spaces are a central step in our analysis.
Our existence result on the problem (1.1) ensures that it admits a solution u, provided that the datum f has a sufficiently small norm in a suitable space modeled on the function
Stronger integrability properties of f are shown to be reflected in a higher degree of integrability of the gradient of u and of the trace of u on
Finally, we prove that weak differentiability of the datum f and its membership in an Orlicz–Sobolev space on
To give the flavor of the conclusions that can be deduced from these results, consider nonlinearities
for some
and has a sufficiently small norm.
Here,
for some
The existence and regularity results about the problem (1.1) outlined above admit analogs for parallel problems for polyharmonic equations. These problems read:
for
Of course, the same kind of problems can be considered in sufficiently regular and bounded open sets instead of half-spaces. Methods similar to those employed in this paper are likely to apply. However, new technical issues have to be faced, since integral operators with more complicated kernels, appearing in representation formulas for solutions in domains, have to be dealt with. On the other hand, dealing with sets of finite measure entails simplifications in the function space setting. A close inspection of these questions does not fall in the scope of our work.
The paper is structured as follows. The next section is devoted to notations and the background from the theory of Young functions and Orlicz and Orlicz–Sobolev spaces. Our main results are stated in Section 3. The subsequent Section 4 contains technical results about Sobolev conjugates of Young functions satisfying specific assumptions. Boundedness properties of some classical operators from harmonic analysis, in Orlicz spaces, are established in Section 5. They are central in view of our approach, which makes substantial use of representation formulas via integral operators. The proofs of the main results about the problems (1.1) and (1.6) are collected in Sections 6 and 7, respectively. Although the problem (1.1) is a special case of (1.6) (the condition on the second line being absent in this case), we prefer to offer a self-contained treatment of the former. It involves simpler notation, and readers who are just interested in second-order equations will find a proof that does not need a detour through the higher-order case.
2 Functional background
2.1 Young functions
The class of Orlicz spaces extends that of Lebesgue spaces in that the role of powers in the definition of the norm is played by more general Young functions.
A
function
for some non-decreasing, left-continuous function
The function
One has that
Moreover,
The Young conjugate
Note the representation formula
where
where
A Young function A is said to satisfy the
for
The function A is said to satisfy the
for
One has that
and
The
Lemma 2.1.
Let A be a Young function having the form (2.1) and let
- (2.11)
- (2.12)
Proof.
Part (i) is well known, and easily verified, thanks to equations (2.1) and (2.3).
As for part (ii), one has that
A Young function A is said to dominate another Young function B globally if there exists a positive constant c such that
for
This terminology and notation will also be adopted for merely nonnegative functions, which are not necessarily Young functions.
The global upper and lower
Matuszewska–Orlicz indices
The indices
The optimal n-dimensional Sobolev conjugate
The function
Let
Then the Sobolev conjugate
In (2.16), and in similar integral conditions in what follows, we indicate just one endpoint in the integral, to emphasize that the condition in question depends on the convergence or divergence of the integral at the relevant endpoint, whereas it is independent of the choice of the other one in
Note that
On the other hand, if
then
whence
for the same values of t.
Finally, we associate with a Young function A the Young function
The Young function
2.2 Function spaces
Let Ω be a measurable set in
finite. In particular,
For some steps of our proofs, we shall need to consider the functional (2.24) associated with a non-decreasing and left-continuous function
The Hölder-type inequality
holds for every
More generally, if A, B are non-decreasing and left-continuous functions, and C is a Young function such that
for some constant k, then
for every
If A dominates B globally, then
for every
Assume now that Ω is an open set and let
Classically,
Here,
The space
The Orlicz–Sobolev space
and is a Banach space endowed with the norm
Here, and in what follows, we use the notation
The homogeneous Orlicz–Sobolev space
and is equipped with the seminorm
In the case when
An embedding theorem for the space
for every function
In particular, if A grows so fast near infinity for the condition (2.20) to be satisfied, then, owing to (2.22),
for some constant
We also need to introduce Orlicz–Sobolev spaces of functions on
In particular, the trace
here, and in what follows,
Given two Young functions A and B, we define the space
endowed with the norm
Higher-order Orlicz–Sobolev spaces come into play to deal with polyharmonic problems. They require the use of spaces defined in terms of k-th order gradients of a function u defined as
Notice that
The homogeneous Orlicz–Sobolev space
and is equipped with the seminorm
In analogy with (2.36), we set
A similar argument as in the first-order case shows that the trace
Furthermore, given two Young functions A and B, we define the space
endowed with the norm
In what follows, when there is no ambiguity, the value of the trace
3 Main results
Our main results admit a unified formulation for the problems (1.6) of arbitrary order m. Because of the importance of second-order problems and of the ease of the corresponding notation, we enucleate the pertaining statements in Section 3.1. The higher-order case is discussed in the subsequent Section 3.2.
3.1 Harmonic functions
In order to formulate the assumptions in our main results we need to introduce some notation and definitions.
Let
and
for some positive constants c and θ. Here, and in similar occurrences in what follows, the function
Next, define the finite-valued Young function E as
where a denotes the function from equation (2.1).
The conditions (2.16) and (2.19) are fulfilled with
Finally, let
The functions
Definition 3.1 (Weak solutions to (1.1)).
Assume that
for every
Our existence result for weak solutions to the problem (1.1) reads as follows.
Theorem 3.2 (Existence of weak solutions to (1.1)).
Assume that
and for some constant
then the problem (1.1) admits a weak solution
Assume, in addition, that
The next theorem amounts to a Sobolev regularity result for the solution u to the problem (1.1) under a stronger integrability assumption on the datum f. It tells us that the membership of f in an Orlicz space whose norm is stronger, in a qualified sense, than
The Orlicz ambient space for the datum f is described via a Young function F such that
The relevant Orlicz space is associated with the Young function
Note that
The Sobolev conjugate
Theorem 3.3 (Sobolev regularity of solutions to (1.1)).
Assume that A, E and
Let
Remark 3.4.
If F is a Young function whose Matuszewska–Orlicz index satisfies
Our last main result about the problem (1.1) proposes additional assumptions
on the data
The function
The function f is assumed to belong to an Orlicz–Sobolev space defined by some Young function M which agrees with E for small values of its argument, and is suitably modified for large values. The function M has to be finite-valued and obeys
Observe that the second condition in (3.12) implies that the Sobolev conjugate
Theorem 3.5 (Smooth solutions to (1.1)).
Let A be as in Theorem 3.2, let D be the function defined by (3.3), and let M be a Young function satisfying (3.12). Assume that:
The function D admits the lower bound
(3.14)for some non-decreasing continuous functions
(3.15)for some positive constants γ,
The function
(3.16)for some positive constants θ and c.
Then there exists a constant
Remark 3.6.
Plainly, only the asymptotic behavior near zero of the functions ϕ and ψ in Theorem 3.5 is relevant.
Remark 3.7.
The condition (3.15) can be equivalently formulated by requiring that the index
Remark 3.8.
In analogy with the inequality (3.2), one may be tempted to simplify the assumption (3.16) as
However, the latter condition is informative only when
In particular, if
Specific choices of the functions ϕ and ψ in (3.14) are described in the examples below.
Example 3.9.
Assume that
for some
Hence, from equations (3.4) and (3.5) one infers that
Moreover, if
whereas, for
Example 3.10.
Assume that
for some
Hence,
As for the condition (3.14), if either
whereas, if either
Example 3.11.
Assume that
for some
Therefore,
If
if
3.2 Polyharmonic functions
The results from the previous subsection admit analogs for the polyharmonic problem (1.6) of arbitrary order
Set
and let
and
for some positive constants c and
and the
Young function
where a denotes the function appearing in (2.1).
Lemma 4.6 ensures that the condition (2.16)
is fulfilled with A replaced with E, and its Sobolev conjugate
The Young function
Definition 3.12 (Weak solutions to (1.6)).
Let
for every
The existence result for solutions to the problem (1.6) is the subject of the following theorem.
Theorem 3.13 (Existence of weak solutions to (1.6)).
Let
then the problem (1.6) admits a weak solution
Assume, in addition, that
The Sobolev regularity of the solution to the problem (1.6) calls into play a Young function F such that
its Sobolev conjugate
Theorem 3.14 (Sobolev regularity of solutions to (1.6)).
Let
Let
We conclude by focusing on the boundary regularity of solutions to the problem (1.6). To this purpose, consider a Young function M as in (3.12). Notice that the convergence of the integral in (3.12) implies that
Theorem 3.15 (Smooth solutions to (1.6)).
Let
The function
Then there exists a constant
4 Auxiliary results on Young functions and their conjugates
This section is devoted to some specific properties of Young functions in connection with their Sobolev conjugates, under the assumptions imposed in our main results.
Throughout the section, we assume that
Lemma 4.1.
Let A be a finite-valued Young function fulfilling the condition (2.16) and let
Assume, in addition, that
Then there exists a constant c such that
Proof.
The property (2.2) implies that
Hence,
an equivalent form of (4.1).
Next, observe that the assumption (4.2)
is equivalent to the fact that for every
Thereby,
(4.6)
for suitable constants c and
whence (4.3) follows. ∎
The statement of the next lemma involves the notion of quasi non-decreasing function. This means that the relevant function fulfills the definition of non-decreasing monotonicity, up to a multiplicative constant.
Lemma 4.2.
Let A be a finite-valued Young function fulfilling the condition (2.16).
Assume that
(4.7)Assume that
(4.8)
Proof.
Part (i). Set
One has that
Since
where a is the function appearing in the representation formula (2.1).
Given
if and only if
On the other hand,
We have that
(4.15)
where the first inequality holds thanks to equation (4.11), and the second one since, by (2.2),
Owing to equation (4.14), an application of the inequality (4.15) with
Part (ii). As noticed above, under the assumption (2.20), one has that
Set
If
Altogether, the property (4.8) follows also in this case. ∎
Let
Moreover,
let
The latter function is equivalent to the original Sobolev conjugate
Lemma 4.3.
Let A be a finite-valued Young function fulfilling the condition (2.16) and let
Proof.
From the inequalities (2.3) one can deduce that
We still need to introduce more
functions associated with
the Young function A. They are the function
and the Young function
Moreover, let E be the function from (3.39), with
Then we denote by
Also, we set
Lemma 4.4.
Let A be a finite-valued Young function satisfying (2.16) and let
Proof.
We start by showing that
is non-decreasing on
and the last expression is nonnegative thanks to the facts that
To show that
Equation (4.28) is an easy consequence of the increasing monotonicity of the function
Lemma 4.5.
Let A be a finite-valued Young function satisfying the condition (3.7). Then the function E defined by (4.23) is a Young function such that
Proof.
A change of variables in the integral on the right-hand side of equation (4.23) yields the alternate formula
Since both a and
Lemma 4.6.
Let A be a finite-valued Young function and let E be the function defined by (4.23). Then
and
Moreover, the Sobolev conjugate
Proof.
Equation (4.29) yields the formula
for the left-continuous derivative of E. Therefore,
The properties (4.30) and (4.31) follow from (4.34), via equation (2.3) applied to E.
Finally, equation (4.32) is a consequence of (4.33) and of Lemma 4.3, applied with A replaced with E. ∎
Lemma 4.7.
Assume that A is a finite-valued Young function
and E is defined as in (4.23). Let
Proof.
An application of Lemma 4.4, with A replaced with E, tells us that
From Lemma 4.3, applied with A replaced with E, we infer that
for some positive constant
Coupling equation (3.44) with (4.37) yields the inequality (4.35). ∎
Lemma 4.8.
Assume that A is a finite-valued Young function.
Let D be defined as in (3.3), let E be defined as in (3.4), let
and
Proof.
From equation (4.20) and the property (2.3), both applied with A replaced with E, one infers that
and
Since
Equation (4.34) tells us that
Combining equations (4.42) and (4.43) yields
Thanks to (4.40) we hence deduce that
namely, (4.38). Similarly, from (4.41) one obtains that
namely, (4.39). ∎
5 Boundedness properties of integral operators in Orlicz spaces
In this section, we establish boundedness properties in Orlicz spaces of the integral operators involved in representation formulas for the solutions to the problems (1.1) and (1.6). Some specific properties of Orlicz norms built upon the Young functions introduced in the previous sections are also collected in the final part of the section.
Let
for
Furthermore, let
for
Also, recall that the classical Riesz potential
for
for
The following properties of the operator
Lemma 5.1.
Let
for
Thanks to this lemma, boundedness properties of any
Lemma 5.2.
Assume that
Namely, there exists a constant c such that
for
The following Sobolev-type boundedness property of the Riesz potential comes into play in the proof of Lemma 5.2.
Lemma 5.3.
Let
Proof.
A proof of equation (5.3) makes use of [17, Theorem 7.2.1, item (ii)], whose proof is, in turn, related to a general characterization of boundedness properties of Riesz potentials in Orlicz spaces from [8]. That theorem enables one to deduce equation (5.3) after verifying that
and
for some constant
and
Equation (5.4) is a consequence of the fact that
which follows from the assumption (2.16) and the property (2.2).
As far as the condition (5.5) is concerned, we observe that, owing to [6, Lemma 1], one has that
for some constant
Thanks to the latter chain, equation (5.6) will follow if we show that there exists a positive constant c such that
Since
This inequality holds as a consequence of the chain
The following lemma ensures that the operator
Lemma 5.4.
Let
Proof.
By the Hölder inequality (2.25),
for
Equation (5.9) thus follows, since the convergence of the last integral is equivalent to the condition (2.16) – see [9, Lemma 2.3]. ∎
Proof of Lemma 5.2.
To begin with, observe that, given a function
for
A parallel bound for the first norm on the left-hand side of the inequality (5.2) relies upon an interpolation argument. Recall that the Poisson extension operator has the following boundedness properties:
see e.g. [13]. From [7, Theorem], one can hence deduce that
provided that there exists a constant c such that
The latter inequality reads
Since
The assumption
see e.g. [15]. From Lemma 5.1 and equations (5.13) and (5.15) one deduces that
for some constants c and
It remains to show that the same inequality continues to hold for any
and
for
for
Now, let
for some constant c.
A generalization of a classical result for Lebesgue spaces ensures that there exists a subsequence, still denoted by
On the other hand,
From (5.21) we hence deduce that
The regularity properties of the Neumann potential operator
Lemma 5.5.
Let M be a Young function as in (3.12), and let
for
Proof.
The definition of the function M guarantees that the assumptions of Lemma 5.3 are fulfilled with A replaced with M. Also, thanks to (3.13),
Let
(5.25)
for a.e.
for suitable constants
Moreover,
for a suitable constant c, and
for
for some constants c and
We have thus shown that
for some constant c and for
Finally, our assumptions on M ensure, via the property (2.35), that
Lemma 5.6.
Let
for
Proof.
Passing to inverse functions in equation (4.28) yields
Consequently, by the first inequality in (4.19),
The inequality (5.26) hence follows as a special case of the inequality (2.27). ∎
Lemma 5.7.
Under the same assumptions as in Lemma 4.7, there exists a constant c such that
for
Lemma 5.8.
Assume that
Proof.
Equation (4.43) tells us that
Thus, thanks to Lemma 4.4, applied with A replaced with E, one has that
Now, choose
This tells us that,
if u is such that
Hence, the inequality (5.31) follows. ∎
6 Proofs of Theorems 3.2, 3.3, and 3.5: Harmonic functions
Our results about the second-order problem (1.1) are established in this section. As explained in Section 1, although they are a special case of those concerning (1.6), a separate proof is offered for the readers’ convenience.
Proof of Theorem 3.2.
Set, for simplicity of notation,
Given a function
for
Note that the last equality holds since, for each given
where
The assertion about the existence of a weak solution to the problem (1.1) will thus follow if we show that,
if
Let
provided that ε is small enough. To verify this fact, fix
From Lemma 5.2 applied with A replaced with E, we obtain
for some constant c independent of u.
Note that the assumptions of Lemma 5.2 are satisfied, with
for some constant
Therefore, if
then equation (6.4) implies that
As a next step, we show that ε can be chosen so small that the operator
(6.7)
for suitable constants c and
whenever
for
Altogether, we have shown that if ε is small enough, then the map (6.2) is a contraction. An application of the Banach fixed point theorem tells us that it admits a unique fixed point u.
The additional piece of information that
for some constant
The inequality (3.8) is consequence of (6.6) and of the fact that
It remains to prove the assertion about the sign of u.
Since the solution u is a fixed point of the operator
If f is positive a.e. and
Proof of Theorem 3.3.
Set, for brevity,
In order to prove that
By (3.1), the definition of D, the assumption (3.10) and the inequality (5.27),
Owing to Lemma 5.8,
thanks to the smallness of
Next, the following chain holds for
for suitable constants c and
We claim that
By Lemma 5.8, this claim follows since, as observed above,
Hence,
Proof of Theorem 3.5.
The solution under consideration to the problem (1.1)
is the limit in
Then
Our goal is to show that
Let
for
Assume that v is any function in
Let
The function
for a sufficiently small
By Lemma 4.8 and Lemma 4.4 applied with A replaced with M, the number
where
Let
where
On the other hand,
Hence,
Notice that
for some constant c,
where the inequality is a consequence of Lemma 5.5. This entails the smallness of
for some
By Lemma 5.5 again,
for some constant c. Moreover,
Assume, in addition, that ε is such that
Therefore,
for some constants c and
Next, equation (3.2) implies that
and for some constant c.
Hence, an analogous chain as in (6.24), with (6.19) now applied with
for some constants c and
provided that ε is sufficiently small, and
The next step consists of a bound for
(6.28)
for sufficiently small ε and for suitable constants
for
An application of Lemma 5.5 tells us that
(6.30)
for some constant
for
Thanks to (6.25) and to analogous steps as above,
(6.33)
for some constants
Finally, the same arguments that
yield (6.19), with
for some constant c.
From the assumption (3.16), the inequality (6.16), the inequality (2.27) in the version under
(6.35)
for suitable constants
for some constant c and for
for some constants
7 Proofs of Theorems 3.13–3.15: Polyharmonic functions
Our approach to the higher-order problem (1.6) is similar to that exposed in the previous section. In the proofs offered below, we shall thus mainly focus on the steps requiring major variants.
Proof of Theorem 3.13.
Consider the operator
for
for every
Hence, if φ is as above, then
An iterated integration by parts and the use of the conditions
(7.4)
Coupling (7.3) with (7.4) produces
As a consequence, thanks to equation (5.20), the following chain holds:
namely, (7.2).
Our aim is now to show that
the integral equation (7.1) admits a unique fixed point u, provided that the norm
To prove that
for sufficiently small ε, one can argue as in the proof of Theorem 3.2, and obtain that
for some constant c and for
whenever
As for the contraction property of the operator
for some constant c and for
provided that
for
Since the kernel
for some constant
Being the fixed point of the map (7.5), the function u is the limit in
If
Proof of Theorem 3.14.
The proof is analogous to that of Theorem 3.3. One has just to define the space
to replace the first-order gradient
Proof of Theorem 3.15.
Set
If
then, by Lemma 5.5,
Consider the sequence
We shall prove that
Thanks to (3.46), there exists a constant
for
Define the functions
Moreover,
by Lemma 4.8 and Lemma 4.4 applied with A replaced with M, there exists
where
Let
where
where
In particular, if
then equation (7.15) combined with (7.13) implies that
From Lemma 5.3, applied with A replaced with M, we deduce that
Next, assume that
for some
We have that
(7.18)
for some constant c. Through an appropriate choice of σ in (7.16), an analogous chain as in (6.24) tells us that there exists a constant c such that
Also, similarly to (6.26),
for some constant c.
Therefore, an induction argument ensures that
for suitable constants
for
Now, observe that there exists a constant
Equations (7.21) and (7.23) imply that
provided that ε is sufficiently small. On the other hand,
The same steps as in (6.33) enable one to deduce that
for sufficiently small ε, where
It remains to estimate the second addend on the right-hand side of (7.26).
Thanks to (7.15), applied with
for some constant c. From (3.16), (7.15) and (7.28) we infer, analogously to (6.35), that
Combining equations (7.24), (7.25), (7.27) and (7.29) ensures that
for some constant c. Iterating this inequality, we obtain, as in (6.36),
for some constant c and for
for some constant c and for
Funding source: Ministero dell’Istruzione, dell’Università e della Ricerca
Award Identifier / Grant number: 201758MTR2
Award Identifier / Grant number: 20229M52AS
Funding statement: This research was partly funded by: (I) GNAMPA of the Italian INdAM – National Institute of High Mathematics (grant number not available) (A. Cianchi); (II) Research Project of the Italian Ministry of Education, University and Research (MIUR) Prin 2017 “Direct and inverse problems for partial differential equations: Theoretical aspects and applications”, grant number 201758MTR2 (A. Cianchi); (III) Research Project of the Italian Ministry of Education, University and Research (MIUR) Prin 2022 “Partial differential equations and related geometric-functional inequalities”, grant number 20229M52AS, co-funded by PNRR (A. Cianchi); (IV) Grant no. 23-04720S of the Czech Science Foundation (L. Slavíková); (V) Primus research programme PRIMUS/21/SCI/002 of Charles University (L. Slavíková); (VI) Charles University Research Centre program No. UNCE/24/SCI/005 (L. Slavíková).
Acknowledgements
We wish to thank the referees for carefully reading the paper and for their valuable comments.
References
[1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. 10.1002/cpa.3160120405Suche in Google Scholar
[2] A. Alberico, A. Cianchi, L. Pick and L. Slavíková, Fractional Orlicz–Sobolev embeddings, J. Math. Pures Appl. (9) 149 (2021), 216–253. 10.1016/j.matpur.2020.12.007Suche in Google Scholar
[3]
D. H. Armitage,
The Neumann problem for a function harmonic in
[4] P. Cherrier, Problèmes de Neumann non linéaires sur les variétés riemanniennes, J. Funct. Anal. 57 (1984), no. 2, 154–206. 10.1016/0022-1236(84)90094-6Suche in Google Scholar
[5] A. Cianchi, A sharp embedding theorem for Orlicz–Sobolev spaces, Indiana Univ. Math. J. 45 (1996), no. 1, 39–65. 10.1512/iumj.1996.45.1958Suche in Google Scholar
[6] A. Cianchi, Boundedness of solutions to variational problems under general growth conditions, Comm. Partial Differential Equations 22 (1997), no. 9–10, 1629–1646. 10.1080/03605309708821313Suche in Google Scholar
[7] A. Cianchi, An optimal interpolation theorem of Marcinkiewicz type in Orlicz spaces, J. Funct. Anal. 153 (1998), no. 2, 357–381. 10.1006/jfan.1997.3193Suche in Google Scholar
[8] A. Cianchi, Strong and weak type inequalities for some classical operators in Orlicz spaces, J. Lond. Math. Soc. (2) 60 (1999), no. 1, 187–202. 10.1112/S0024610799007711Suche in Google Scholar
[9] A. Cianchi, Optimal Orlicz–Sobolev embeddings, Rev. Mat. Iberoam. 20 (2004), no. 2, 427–474. 10.4171/rmi/396Suche in Google Scholar
[10] A. Cianchi, Higher-order Sobolev and Poincaré inequalities in Orlicz spaces, Forum Math. 18 (2006), no. 5, 745–767. 10.1515/FORUM.2006.037Suche in Google Scholar
[11] M. F. de Almeida and L. S. M. Lima, Nonlinear boundary problem for harmonic functions in higher dimensional Euclidean, preprint (2021), https://arxiv.org/abs/1807.04122v4. Suche in Google Scholar
[12] L. C. F. Ferreira, E. S. Medeiros and M. Montenegro, On the Laplace equation with a supercritical nonlinear Robin boundary condition in the half-space, Calc. Var. Partial Differential Equations 47 (2013), no. 3–4, 667–682. 10.1007/s00526-012-0531-2Suche in Google Scholar
[13] F. Hang, X. Wang and X. Yan, Sharp integral inequalities for harmonic functions, Comm. Pure Appl. Math. 61 (2008), no. 1, 54–95. 10.1002/cpa.20193Suche in Google Scholar
[14] B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential Integral Equations 7 (1994), no. 2, 301–313. 10.57262/die/1369330430Suche in Google Scholar
[15] V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific, River Edge, 2008. Suche in Google Scholar
[16] K. Medville and M. S. Vogelius, Existence and blow up of solutions to certain classes of two-dimensional nonlinear Neumann problems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 23 (2006), no. 4, 499–538. 10.1016/j.anihpc.2005.02.008Suche in Google Scholar
[17] V. Musil, Classical operators of harmonic analysis in Orlicz spaces, Doctoral Thesis, Charles University, Prague, 2018. Suche in Google Scholar
[18] C. B. Ndiaye, Constant T-curvature conformal metrics on 4-manifolds with boundary, Pacific J. Math. 240 (2009), no. 1, 151–184. 10.2140/pjm.2009.240.151Suche in Google Scholar
[19] R. O’Neil, Fractional integration in Orlicz spaces. I, Trans. Amer. Math. Soc. 115 (1965), 300–328. 10.1090/S0002-9947-1965-0194881-0Suche in Google Scholar
[20] P. Quittner and W. Reichel, Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions, Calc. Var. Partial Differential Equations 32 (2008), no. 4, 429–452. 10.1007/s00526-007-0155-0Suche in Google Scholar
[21] L. Sun and J. Xiong, Classification theorems for solutions of higher order boundary conformally invariant problems, I, J. Funct. Anal. 271 (2016), no. 12, 3727–3764. 10.1016/j.jfa.2016.08.020Suche in Google Scholar
[22] M. Vogelius and J.-M. Xu, A nonlinear elliptic boundary value problem related to corrosion modeling, Quart. Appl. Math. 56 (1998), no. 3, 479–505. 10.1090/qam/1637048Suche in Google Scholar
[23] G. D. Yomgne, Nonlinear biharmonic equation in half-space with rough Neumann boundary data and potentials, Nonlinear Anal. 215 (2022), Article ID 112623. 10.1016/j.na.2021.112623Suche in Google Scholar
[24] G. D. Yomgne, On a nonlinear Laplace equation related to the boundary Yamabe problem in the upper-half space, Commun. Pure Appl. Anal. 21 (2022), no. 2, 517–539. 10.3934/cpaa.2021186Suche in Google Scholar
[25]
L. Zhang,
Classification of conformal metrics on
© 2025 Walter de Gruyter GmbH, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 International License.
Artikel in diesem Heft
- Frontmatter
- A characterization of ℓ1 double bubbles with general interface interaction
- The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
- On the behavior in time of the solutions to total variation flow
- Zaremba problem with degenerate weights
- Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity
- Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group
- On the variational nature of the Anzellotti pairing
- Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space
- Upper semicontinuity of index plus nullity for minimal and CMC hypersurfaces
- A variational approach to the Navier–Stokes equations with shear-dependent viscosity
- Hölder regularity for the fractional p-Laplacian, revisited
- Poincaré inequality and energy of separating sets
- On a class of obstacle problems with (p, q)-growth and explicit u-dependence
- Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups
- Parabolic Lipschitz truncation for multi-phase problems: The degenerate case
Artikel in diesem Heft
- Frontmatter
- A characterization of ℓ1 double bubbles with general interface interaction
- The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
- On the behavior in time of the solutions to total variation flow
- Zaremba problem with degenerate weights
- Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity
- Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group
- On the variational nature of the Anzellotti pairing
- Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space
- Upper semicontinuity of index plus nullity for minimal and CMC hypersurfaces
- A variational approach to the Navier–Stokes equations with shear-dependent viscosity
- Hölder regularity for the fractional p-Laplacian, revisited
- Poincaré inequality and energy of separating sets
- On a class of obstacle problems with (p, q)-growth and explicit u-dependence
- Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups
- Parabolic Lipschitz truncation for multi-phase problems: The degenerate case
