Gradient potential estimates for elliptic obstacle problems

doi:10.1016/j.jmaa.2020.124698

Journal of Mathematical Analysis and Applications

Volume 495, Issue 1, 1 March 2021, 124698

https://doi.org/10.1016/j.jmaa.2020.124698 Get rights and content

Abstract

In this paper, we consider the solutions of the non-homogeneous quasilinear elliptic obstacle problems involving measure data. At first we prove the existence of approximable solutions to these problems. Then we establish pointwise and oscillation estimates for the gradients of solutions via the nonlinear Wolff potentials, and these yield results on $C^{1, α}$ -regularity of solutions.

Section snippets

Introduction and main results

The obstacle problems discussed in this paper are related to the following quasilinear equation: $div (a (| D u |) D u) = μ in Ω$ where $Ω \subseteq R^{n}, n ⩾ 2$ is a bounded open set and μ is a Radon measure defined on Ω with finite total mass. Moreover we assume that $μ (R^{n} \ Ω) = 0$ and $a : (0, \infty) \to (0, \infty) \in C^{1} (0, \infty)$ satisfies $0 \leq i_{a} = : \inf_{t > 0} \frac{t a^{'} (t)}{a (t)} \leq \sup_{t > 0} \frac{t a^{'} (t)}{a (t)} = : s_{a} < \infty .$ The obstacle condition that we impose on the solution is of the form $u \geq ψ$ a.e. on Ω, where $ψ \in W^{1, G} (Ω) \cap W^{2, 1} (Ω)$ is a given function with $d i v [a (| D ψ |) D ψ] \in L^{1} (Ω)$ and G is

Preliminaries

In this section, we introduce some notions and results which will be used in this paper. Firstly, we denote by m any number in the natural number set $N$ , it is easily verified that ${∥ f - {(f)}_{Ω} ∥}_{L^{2} (Ω)} = \min_{c \in R^{m}} {∥ f - c ∥}_{L^{2} (Ω)}$ for any measurable set $Ω \subseteq R^{n}$ and every function $f : Ω \to R^{m}$ such that $f \in L^{2} (Ω)$ . If $q \in [1, \infty]$ , we have ${∥ f - {(f)}_{Ω} ∥}_{L^{q} (Ω)} ⩽ 2 \min_{c \in R^{m}} {∥ f - c ∥}_{L^{q} (Ω)} .$

Definition 2.1

A Young function B is called an N-function if $0 < B (t) < + \infty f o r t > 0$ and $\lim_{t \to + \infty} \frac{B (t)}{t} = \lim_{t \to 0} \frac{t}{B (t)} = + \infty .$ The Young conjugate $B^{⁎}$ of a Young function B is defined as $B^{⁎} = \sup_{s \geq}$

Existence result

In this section we shall prove the existence result for the obstacle problems with measure data. Let's start with the following lemma, which will be useful for Lemma 3.2.

Lemma 3.1

Let $Ω \subset R$ be a bounded domain. Assume that $2 + i_{a} ⩽ n$ , $h \in T_{0}^{1, G} (Ω)$ satisfies $\int_{Ω \cap {| h | ⩽ k}} | D h |^{2 + i_{a}} d x ⩽ M k + M^{\frac{2 + i_{a}}{1 + i_{a}}}$ for $\forall k > 0$ , and fixed constants $M > 0$ . Then we have $\int_{Ω} | h |^{1 + α} d x ⩽ c_{1} M^{\frac{1 + α}{1 + i_{a}}},$ $\int_{Ω} | D h |^{1 + β} d x ⩽ c_{2} M^{\frac{1 + β}{1 + i_{a}}},$ where $0 < α < m i n {1, \frac{n i_{a} + 2 + i_{a}}{n - 2 - i_{a}}}, 0 < β < m i n {1, \frac{n i_{a} + 1}{n - 1}}, c_{1} = c_{1} (Ω, n, i_{a}, α), c_{2} = c_{2} (Ω, n, i_{a}, β)$ .

Proof

We first assume that $M = 1$ , the inequality (3.1)

Comparison estimates

This section is devoted to compare the solutions of obstacle problems to the solutions of elliptic equations. Therefore we can obtain a similar excess decay estimate for solutions of obstacle problems with measure data.

Firstly, we shall prove the comparison estimate between the solutions of elliptic obstacle problems with measure data and the solutions of the corresponding homogeneous obstacle problems.

Lemma 4.1

Under the assumption (1.2), let $B_{2 R} (x_{0}) \subset Ω, f \in L^{1} (B_{R} (x_{0})) \cap {(W^{1, G} (B_{R} (x_{0})))}^{'}$ and the map $u \in W^{1, G} (B_{R} ($

Gradient estimates by potentials

In this section, we first prove a pointwise estimate of fractional operator by precise iteration methods, and therefore allow to get $L^{\infty}$ -bounds. We thus obtain pointwise and oscillation estimates for the gradients of solutions by means of the sharp maximal function estimates.

Proof of Theorem 1.7

We choose a geometric sequence $R_{i}$ whose spread $H > 1$ will be a certain function of the fixed quantities $n, i_{a}, s_{a}$ , and will be determined later. More precisely we define $B_{i} : = B (x, \frac{R}{H^{i}}) = B (x, R_{i}), for i = 0, 1, 2, . . .,$ ${\hat{A}}_{i} : = R_{i}^{- α} \underset{B_{i}}{⨏} | D u - {(D u)}_{B_{i}} |$

Acknowledgments

The authors are supported by the National Natural Science Foundation of China (NNSF Grant No.12071229 and No.11671414). The authors would like to express their gratitude to the anonymous reviewers for their constructive comments and suggestions that improved the last version of the manuscript.

References (36)

L. Boccardo et al.
Nonlinear elliptic and parabolic equations involving measure data
J. Funct. Anal.
(1989)
A. Cianchi et al.
Quasilinear elliptic problems with general growth and merely integrable, or measure, data
Nonlinear Anal.
(2017)
F. Duzaar et al.
Gradient estimates via linear and nonlinear potentials
J. Funct. Anal.
(2010)
T. Kuusi et al.
Universal potential estimates
J. Funct. Anal.
(2012)
L. Liu et al.
Restricting Riesz-Morrey-Hardy potentials
J. Differ. Equ.
(2017)
P. Oppezzi et al.
Unilateral problems with measure data
Nonlinear Anal.
(2001)
C. Scheven
Gradient potential estimates in non-linear elliptic obstacle problems with measure data
J. Funct. Anal.
(2012)
F.P. Yao
Calderón-Zygmund estimates for a class of quasilinear parabolic equations
Nonlinear Anal.
(2018)
F.P. Yao et al.
Gradient estimates via the Wolff potentials for a class of quasilinear elliptic equations
J. Math. Anal. Appl.
(2017)
P. Baroni
Riesz potential estimates for a general class of quasilinear equations
Calc. Var. Partial Differ. Equ.
(2015)

L. Beck et al.

Lipschitz bounds and nonuniform ellipticity

Commun. Pure Appl. Math.

(2020)

L. Boccardo et al.

Nonlinear elliptic equations with right-hand side measures

Commun. Partial Differ. Equ.

(1992)

S. Challal et al.

On the A-obstacle problem and the Hausdorff measure of its free boundary

Ann. Mat. Pura Appl. (4)

(2012)

A. Cianchi et al.

Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems

J. Eur. Math. Soc.

(2014)

A. Cianchi et al.

Global Lipschitz regularity for a class of quasilinear elliptic equations

Commun. Partial Differ. Equ.

(2011)

A. Cianchi et al.

Global boundedness of the gradient for a class of nonlinear elliptic systems

Arch. Ration. Mech. Anal.

(2014)

F. Duzaar et al.

Gradient estimates via non-linear potentials

Am. J. Math.

(2011)

L. Hedberg et al.

Thin sets in nonlinear potential theory

Ann. Inst. Fourier (Grenoble)

(1983)

Cited by (9)

Global gradient estimates for general nonlinear elliptic measure data problems with Orlicz growth
2023, Journal of Mathematical Analysis and Applications
We provide an optimal global Calderón-Zygmund theory for gradients of SOLAs to general nonlinear elliptic equations $- div A (x, u, D u) = μ$ whose principle part depends on the solution itself and right-hand data μ is a signed Radon measure. The associated nonlinearity $A$ is assumed to satisfy the $(δ, R_{0})$ -BMO condition in x, local uniform continuity in u, and Orlicz growth condition in Du, while the boundary of underlying domain is assumed to be Reifenberg flat. This is achieved by employing a perturbation method together with developing a one-parameter technique and by applying the maximal function free technique.
Riesz potential estimates for problems with Orlicz growth
2022, Journal of Mathematical Analysis and Applications
Citation Excerpt :
As for the elliptic equations with Orlicz growth, Baroni established Riesz potential estimates for gradient of solutions to elliptic equations with constant coefficients in [2]. Later, pointwise and oscillation estimates for the gradient of solutions via Wolff potentials were further extended to elliptic obstacle problems by Xiong, Zhang and Ma [33,34]. Our goal in this work is to extend the Riesz potential estimates in [2] to elliptic equations with Dini-BMO coefficients; as a result, the potential estimates open the way to providing a sharp continuity criterium for the gradient.
In this paper, we consider the solutions of the non-homogeneous quasilinear elliptic equations with Dini-BMO coefficients involving measure data. At first, we prove pointwise gradient estimates for solutions by Riesz potentials; as a consequence, we obtain the borderline gradient regularity and extend a classical theorem of Stein for Poisson equations. Then we establish oscillation estimates of solutions via Riesz potentials, and these yield Hölder continuity of solutions.
Toward Weighted Lorentz–Sobolev Capacities from Caffarelli–Silvestre Extensions
2024, Journal of Geometric Analysis
Nonlinear Potential Estimates for Generalized Stokes System
2022, Mediterranean Journal of Mathematics
Gradient potential estimates in elliptic obstacle problems with Orlicz growth
2022, Calculus of Variations and Partial Differential Equations
The Poisson equation involving surface measures
2022, Communications in Partial Differential Equations

View all citing articles on Scopus

View full text

Gradient potential estimates for elliptic obstacle problems

Abstract

Section snippets

Introduction and main results

Preliminaries

Existence result

Comparison estimates

Gradient estimates by potentials

Acknowledgments

J. Funct. Anal.

Nonlinear Anal.

J. Funct. Anal.

J. Funct. Anal.

J. Differ. Equ.

Nonlinear Anal.

J. Funct. Anal.

Nonlinear Anal.

J. Math. Anal. Appl.

Riesz potential estimates for a general class of quasilinear equations

Calc. Var. Partial Differ. Equ.

Lipschitz bounds and nonuniform ellipticity

Commun. Pure Appl. Math.

Nonlinear elliptic equations with right-hand side measures

Commun. Partial Differ. Equ.

On the A-obstacle problem and the Hausdorff measure of its free boundary

Ann. Mat. Pura Appl. (4)

Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems

J. Eur. Math. Soc.

Global Lipschitz regularity for a class of quasilinear elliptic equations

Commun. Partial Differ. Equ.

Global boundedness of the gradient for a class of nonlinear elliptic systems

Arch. Ration. Mech. Anal.

Gradient estimates via non-linear potentials

Am. J. Math.

Thin sets in nonlinear potential theory

Ann. Inst. Fourier (Grenoble)