Wolff Potentials Research Articles

Wolff potential estimates and Wiener criterion for nonlocal equations with Orlicz growth

We prove the Wolff potential estimates for nonlocal equations with Orlicz growth. As an application, we obtain the Wiener criterion in this framework, which provides a necessary and sufficient condition for boundary points to be regular. Our approach relies on the fine analysis of superharmonic functions in view of nonlocal nonlinear potential theory.

Decay estimates for quasilinear elliptic equations and a Brezis–Nirenberg result in D1,p(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^{1,p}({\\mathbb R}^N)$$\\end{document}

We prove decay estimates for solutions of quasilinear elliptic equations in the whole RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb R}^N$$\\end{document} of the form u∈X:-divA(x,∇u)=a(x)f(x,u),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u\\in X: -\ ext{ div }\\,A(x,\ abla u)=a(x) f(x,u), \\end{aligned}$$\\end{document}where X=D1,p(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=D^{1,p}({\\mathbb R}^N)$$\\end{document} is the Beppo-Levi space (also called homogeneous Sobolev space). Based on these decay estimates we are able to prove a Brezis–Nirenberg type result for the energy functional Φ:X→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi : X\\rightarrow {\\mathbb R}$$\\end{document} related to the p-Laplacian equation in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb R}^N$$\\end{document} in the form u∈X:-Δpu=a(x)g(u),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u\\in X: -\\Delta _p u=a(x) g(u), \\end{aligned}$$\\end{document}saying that for the subspace V of bounded continuous functions with weight 1+|x|N-pp,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1+|x|^{\\frac{N-p}{p}},$$\\end{document} a local minimizer of Φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi $$\\end{document} in the finer V topology is also a local minimizer in the X-topology. Global L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document}-estimates on the one hand and pointwise estimates for solutions of quasilinear elliptic equations in terms of nonlinear Wolff potentials on the other hand play a crucial role in the proofs.

Singular Yamabe problem for scalar flat metrics on the sphere

Let Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} be a domain on the unit n-sphere Sn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb {S}}^n$$\\end{document} and g∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overset{{\\,}_\\circ }{g}$$\\end{document} the standard metric of Sn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {S}}^n$$\\end{document}, n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 3$$\\end{document}. We show that there exists a conformal metric g with vanishing scalar curvature R(g)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R(g)=0$$\\end{document} such that (Ω,g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Omega , g)$$\\end{document} is complete if and only if the Bessel capacity Cα,q(Sn\\Ω)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}_{\\alpha , q}({\\mathbb {S}}^n\\setminus \\Omega )=0$$\\end{document}, where α=1+2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =1+\\frac{2}{n}$$\\end{document} and q=n2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=\\frac{n}{2}$$\\end{document}. Our analysis utilizes some well known properties of capacity and Wolff potentials, as well as a version of the Hopf–Rinow theorem for the divergent curves.

Wolff potentials and local behavior of solutions to elliptic problems with Orlicz growth and measure data

Abstract We establish pointwise bounds expressed in terms of a nonlinear potential of a generalized Wolff type for 𝒜 {{\mathcal{A}}} -superharmonic functions with nonlinear operator 𝒜 : Ω × ℝ n → ℝ n {{\mathcal{A}}:\Omega\times{\mathbb{R}^{n}}\to{\mathbb{R}^{n}}} having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls estimates from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfy conditions expressed in the natural scales. Finally, we give a variant of Hedberg–Wolff theorem on characterization of the dual of the Orlicz space.

Pointwise estimates of solutions to weighted parabolic p-Laplacian equation via Wolff potential

For the weighted parabolic equation \v\left(x \right)u_{t} -{\hbox{div}({w(x)| \nabla u |^{p-2}}} \nabla u) = f , p >{2} we prove the local boundedness for weak solutions in terms of the weighted Wolff potential of the right-hand side of equation.

Wolff potentials and measure data vectorial problems with Orlicz growth

We study solutions to measure data elliptic systems with Uhlenbeck-type structure that involve operator of divergence form, depending continuously on the spacial variable, and exposing doubling Orlicz growth with respect to the second variable. Pointwise estimates for the solutions that we provide are expressed in terms of a nonlinear potential of generalized Wolff type. Not only we retrieve the recent sharp results proven for p-Laplace systems, but additionally our study covers the natural scope of operators with similar structure and natural class of Orlicz growth.

POINTWISE ESTIMATES OF SOLUTIONS TO WEIGHTED POROUS MEDIUM AND FAST DIFFUSION EQUATIONS VIA WEIGHTED RIESZ POTENTIALS

Problems related to the study of the properties of solutions of partial differential equations have attracted the attention of many authors in recent decades. The main qualitative properties of solutions of homogeneous linear elliptic equations of the second-order divergent type with measurable coefficients without lower-order terms are already known from the results of De Giorgi, Nash, and Moser. These results are generalized by Serrin, Ladyzhenska and Uraltseva, Aronson and Serrin, and Trudinger for wide classes of elliptic and parabolic equations with lower-order terms from the corresponding $ L^{q} $-classes. Analogous results for evolution equations with $ p \,-$Laplacian appeared much later. The first significant transition to the $ p \,-$Laplace equation with the measure $~\mu~$ in the right-hand side was achieved by Kilpelainen and Maly. They established point estimates of the solutions in terms of the nonlinear Wolff potential. These results were later extended by \linebreak Trudinger and Wang and Laboutin to nonlinear and subelliptic quasilinear equations. Irregularly elliptic and inhomogeneous parabolic equations without/or with singular lower terms have been studied for a long time. The first results in this direction were obtained by Fabes, Kenig and Separioni and Gutierrez for a weighted linear elliptic equation with weight representing $ A_{2} $ of the Mackenhaupt class. In recent decades, there has been a growing interest in parabolic and elliptic equations due to their application in modeling nonlinear physical processes occurring in heterogeneous media. Also, these equations are interesting because a general qualitative theory has not been constructed for them. Among the researchers who obtained the first significant results, we note Di Benedetto E., Bogelein V., Ivanov A. V., Duzaar F., Gianazza U., Vespri V..

Gradient potential estimates in elliptic obstacle problems with Orlicz growth

In this paper,we consider the solutions of the non-homogeneous elliptic obstacle problems with Orlicz growth involving measure data. We first establish the pointwise estimates of the approximable solutions to these problems via fractional maximal operators. Furthermore, we obtain pointwise and oscillation estimates for the gradients of solutions by the non-linear Wolff potentials, and these yield results on \(C^{1,\alpha }\)-regularity of solutions.

Embeddings of function spaces via the Caffarelli–Silvestre extension, capacities and Wolff potentials

Let Pαf(x,t) be the Caffarelli–Silvestre extension of a smooth function f(x):Rn→R+n+1≔Rn×(0,∞). The purpose of this article is twofold. Firstly, we want to characterize a nonnegative measure μ on R+n+1 such that f(x)→Pαf(x,t) induces bounded embeddings from the Lebesgue spaces Lp(Rn) to the Lq(R+n+1,μ). On one hand, these embeddings will be characterized by using a newly introduced Lp−capacity associated with the Caffarelli–Silvestre extension. In doing so, the mixed norm estimates of Pαf(x,t), the dual form of the Lp−capacity, the Lp−capacity of general balls, and a capacitary strong type inequality will be established, respectively. On the other hand, when p>q>1, these embeddings will also be characterized in terms of the Hedberg–Wolff potential of μ. Secondly, we characterize a nonnegative measure μ on R+n+1 such that f(x)→Pαf(x,t) induces bounded embeddings from the homogeneous Sobolev spaces Ẇβ,p(Rn) to the Lq(R+n+1,μ) in terms of the fractional perimeter of open sets for endpoint cases and the fractional capacity for general cases.

On n-superharmonic functions and some geometric applications

In this paper we study asymptotic behaviors of n-superharmonic functions at singularity using the Wolff potential and capacity estimates in nonlinear potential theory. Our results are inspired by and extend [6] of Arsove–Huber and [63] of Taliaferro in 2 dimensions. To study n-superharmonic functions we use a new notion of thinness in terms of n-capacity motivated by a type of Wiener criterion in [6]. To extend [63], we employ the Adams–Moser–Trudinger’s type inequality for the Wolff potential, which is inspired by the inequality used in [15] of Brezis–Merle. For geometric applications, we study the asymptotic end behaviors of complete conformally flat manifolds as well as complete properly embedded hypersurfaces in hyperbolic space. These geometric applications seem to elevate the importance of n-Laplace equations and make a closer tie to the classic analysis developed in conformal geometry in general dimensions.

Integrability of solutions for the integral system

This paper is concerned with the following integral system involving the Riesz potential and the Wolff potential(1){u(x)=Wβ,γ(up−1v)(x),x∈Rn,v(x)=Iα(up)(x),x∈Rn, where p>1, 1<γ≤2, α∈(0,n) and γβ∈(0,n). By the Hardy-Littlewood-Sobolev inequality and the regularity lifting lemma, we obtain the optimal integrability of integrable solutions (u,v). This integrability is the key ingredient to study the decay rate at infinity of the solutions.

Optimal integrability for some integral system of Wolff type

In the paper, we obtain the optimal integrability for positive solutions of the following integral system involving Wolff potentials: {u(x) = W?,?(vq)(x); x ?, Rn, v(x) = W?,?(up)(x), x ? Rn, where p,q &gt; 0,? &gt; 0, ? &gt; 1 and 0 &lt; ?? &lt; n. Ma, Chen and Li [Advances in Mathematics, 226(2011), 2676-2699] developed the regularity lifting method and obtained the optimal integrability for p &gt; 1, q &gt; 1. Here, based on some new observations, we overcome the difficulty there, and derive the optimal integrability for the case of p &gt; 0,q &gt; 0 and pq &gt; 1. This integrability plays a key role in estimating the asymptotic behavior of positive solutions.

Radial symmetry for a weighted integral equation of Wolff type

In this paper, we consider a weighted integral equation involving Wolff potentials in Rnu(x)=Wβ,p(|y|auq)(x)and a related differential form −div(|∇u|p−2∇u)=|x|auq(x).where p∈1,2, q>p−1, β>0 and 0≤−a<pβ<n. By applying the method of moving plane introduced by Li and Chen (Chen et al., 2006) we are able to prove the symmetry of the positive solutions.

Gradient potential estimates for elliptic obstacle problems

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 C1,α-regularity of solutions.

Quasilinear elliptic equations with a source reaction term involving the function and its gradient and measure data

We study the equation $$-\text {div}(A(x,\nabla u))=|u|^{q_1-1}u|\nabla u|^{q_2}+\mu $$ where $$A(x,\nabla u)\sim |\nabla u|^{p-2}\nabla u$$ in some suitable sense, $$\mu $$ is a measure and $$q_1$$ , $$q_2$$ are nonnegative real numbers and satisfy $$q_1+q_2>p-1$$ . We give sufficient conditions for existence of solutions expressed in terms of the Wolff potential or the Riesz potentials of the measure. Finally we connect the potential estimates on the measure with Lipchitz estimates with respect to some Bessel or Riesz capacity.

Quasilinear elliptic equations with exponential nonlinearity and measure data

Let be either a bounded domain or the entire space. This paper concerns existence and global pointwise estimates in terms of Wolff potentials of solutions to quasilinear and Hessian equations of Lane‐Emden type: under minimal assumptions on σ and ω, where σ is an A∞ weight and ω is a nonnegative Borel measure, and with γ&gt;0, β ≥ 1. Our methods are based on systematic application of Wolff potentials and good‐λ inequality for fractional maximal potential.

Continuity of weak solutions to nonlinear fourth-order equations with strengthened ellipticity via Wolff potentials

In the present article nonlinear fourth-order equations in the divergence form with L^1-right-hand sides and the strengthened ellipticity condition on the coefficients are analyzed. Such equations, but with sufficiently regular right-hand sides, first appeared in the works of Professor I.V. Skrypnik concerning the regularity of generalized solutions for multidimensional nonlinear elliptic equations of high order. This class of equations correctly generalizes the corresponding nonlinear second-order elliptic equations with non-standard growth conditions on the coefficients, which are models for numerous physical phenomena in non-homogeneous medium. The main result of the article is a theorem on an estimation of oscillations in a ball of solutions to the given equations via the Wolff potentials of their right-hand sides. To prove this, we use the improved Kilpelainen-Maly method and pointwise potential estimates of functions related to special subclasses of Sobolev spaces, akin to the well-known De Giorgi classes. A new point is the verification that these classes contain superpositions of solutions and Moser logarithmic functions that include the Wolf potential of the right-hand side of the equation. As a corollary, a new result is obtained on the interior continuity of solutions to the equations with right-hand sides from the Kato class, which is characterized by the uniform convergence to zero of the corresponding Wolff potentials. Some important cases of fulfilling this condition are considered: the right-hand side of the equation belongs to the Morrey space with an index exceeding a certain limiting value, then the solutions are locally Holder continuous; if the right-hand side belongs to the borderline Lorentz-Zygmund classes, then the solutions are only locally continuous, but they are not Holder continuous in the domain. In the case when the summability exponents of the right-hand sides of the equations under consideration are less than the borderline values, there are examples of unbounded discontinuous solutions. These facts are exact analogues of the corresponding results in the theory of second-order elliptic equations.

On the generalized [formula omitted] classes of De Giorgi-Ladyzhenskaya-Ural'tseva and pointwise estimates of solutions to high-order elliptic equations via Wolff potentials

We consider quasilinear elliptic 2m-order (m⩾2) partial differential equations which prototype is∑|α|=m(−1)|α|Dα(|Dmu|p−2Dαu)=f(x),x∈Ω, where Ω is a bounded open set in Rn, n=mp and f∈L1(Ω). Using an analogue of the Kilpeläinen-Malý method, we obtain the local boundedness and continuity of arbitrary weak solution u∈Wm,p(Ω) via the Wolff potential Wm,pf.

Wolff potentials and regularity of solutions to integral systems on spaces of homogeneous type

Two objectives are accomplished in this paper: First we establish the comparison between Wolff and Riesz potentials on space of homogeneous type in the sense of Coifman and Weiss ( Theorem 1.2 ), followed by a Hardy–Littlewood–Sobolev type inequality for Wolff potentials ( Theorem 1.3 ). Then applying this inequality, we consider a Lane–Emden type integral system and use regularity lifting to derive integrability estimates of positive solutions to the system, by which we also imply L ∞ estimates (see Theorem 1.4 , Theorem 1.5 ). Furthermore, we use a modified regularity lifting method to prove that the positive solutions are also Lipschitz continuous ( Theorem 1.6 ). Our results imply Wolff potentials and regularity of solutions to the Lane–Emden type integral systems on stratified groups and the Carnot–Carathedory metric spaces defined by a family of vector fields satisfying Hörmander's finite rank condition as special cases.

Decay estimates for Wolff potentials in [formula omitted] and gradient-dependent quasilinear elliptic equations

We prove decay estimates of positive weak solutions u∈D1,p(RN)∩C1(RN) and their gradients |∇u| of the p-Laplacian problem (1<p<N):u∈D1,p(RN):−Δpu=a(x), where the measurable function a satisfies the decay estimate (α>0):0<a(x)≲11+|x|N+α,∀x∈RN. The positive Borel measure on RN generated by the right-hand side function a and its associated Wolff potential is known to be the main tool for pointwise estimates of u(x)>0 and |∇u(x)|. Therefore, one of the main goal of this paper is to prove decay estimates for corresponding Wolff potentials in RN. The obtained decay estimates in conjunction with a sub-supersolution method, which is developed here for quasilinear elliptic equations, are then applied to prove the existence of positive solutions for classes of gradient-dependent quasilinear equations in the formu∈D1,p(RN):−Δpu=a(x)f(u,∇u).