Superharmonic functions are locally renormalized solutions

2011

Abstract We show that different notions of solutions to measure data problems involving p-Laplace type operators and nonnegative source measures are locally essentially equivalent. As an application we characterize singular solutions of multidimensional Riccati type partial differential equations.

Maximal regularity via reverse Hölder inequalities for elliptic systems of n-Laplace type involving measures

2008

In this note, we consider the regularity of solutions of the nonlinear elliptic systems of n-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space Ln,∞. We also obtain the a priori global and local estimates for the Ln,∞-norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.

Generalized dirichlet problem in nonlinear potential theory

1990

The operator extending the classical solution of the Dirichlet problem for the quasilinear elliptic equation divA(x,▽u)=0 akin to thep-Laplace equation is shown to be unique providedA obeys a specific order principle. The Keldych lemma is also generalized to this nonlinear setting.

The Wiener test and potential estimates for quasilinear elliptic equations

1994

H�lder continuity of solutions to quasilinear elliptic equations involving measures

1994

We show that the solutionu of the equation $$ - div(|\nabla u|^{p - 2} \nabla u) = \mu $$ is locally β-Holder continuous provided that the measure μ satisfies the condition μ(B(x,r))⩽Mrn − p + α(p − 1) for some α>β. A corresponding result for more general operators is also proven.

Lattice property of $p$-admissible weights

2015

p-Laplacian type equations involving measures

2003

This is a survey on problems involving equations $-\operatorname{div}{\Cal A}(x,\nabla u)=\mu$, where $\mu$ is a Radon measure and ${\Cal A}:\bold {R}^n\times\bold R^n\to \bold R^n$ verifies Leray-Lions type conditions. We shall discuss a potential theoretic approach when the measure is nonnegative. Existence and uniqueness, and different concepts of solutions are discussed for general signed measures.

Pointwise regularity of solutions to nonlinear double obstacle problems

1991

On the Porosity of Free Boundaries in Degenerate Variational Inequalities

2000

Abstract In this note we consider a certain degenerate variational problem with constraint identically zero. The exact growth of the solution near the free boundary is established. A consequence of this is that the free boundary is porous and therefore its Hausdorff dimension is less than N and hence it is of Lebesgue measure zero.

BLD -mappings in $W^{2,2}$ are locally invertible

2000

We prove that mappings of bounded length distortion are local homeomorphisms if they have L 2 -integrable weak second derivatives.

Singular solutions to p-Laplacian type equations

1999

We construct singular solutions to equations $div\mathcal{A}(x,\nabla u) = 0,$ similar to the p-Laplacian, that tend to ∞ on a given closed set of p-capacity zero. Moreover, we show that every Gδ-set of vanishing p-capacity is the infinity set of some A-superharmonic function.

Harmonic morphisms in nonlinear potential theory

1992

This article concerns the following problem: given a family of partial differential operators with similar structure and given a continuous mapping f from an open set Ω in Rn into Rn, then when does f pull back the solutions of one equation in the family to solutions of another equation in that family? This problem is typical in the theory of differential equations when one wants to use a coordinate change to study solutions in a different environment.

Nonlinear Potential Theory and PDEs

1994

We consider equations like — div(∣∇u∣ p-2∇u) = µ, where µ is a nonnegative Radon measure and 1 < p < ∞. Results that relate the solution u and the measure µ are reviewed. A link between potential estimates and the boundary regularity of the Dirichlet problem is established.

On the fusion problem for degenerate elliptic equations

1995

Let F be a relatively closed subset of a Euclidean domain Ω. We investigate when solutions u to certain elliptic equations on Ω/F are restrictions of solutions on all of Ω. Specifically, we show that if ∂F is not too large, and u has a suitable decay rate near F, then u can be so extended.

A-superharmonic functions and supersolutions of degenerate elliptic equations

1988

Removable sets for continuous solutions of quasilinear elliptic equations

2001

We show that sets of n − p + α ( p − 1 ) n-p+\alpha (p-1) Hausdorff measure zero are removable for α \alpha -Hölder continuous solutions to quasilinear elliptic equations similar to the p p -Laplacian. The result is optimal. We also treat larger sets in terms of a growth condition. In particular, our results apply to quasiregular mappings.