Dimension of Elliptic Harmonic Measure of Snowspheres

A two-phase free boundary problem for a semilinear elliptic equation

In this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $Dsubset mathbb{R}^{n}$ with smooth boundary‎. ‎We give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of Caffarelli and Friedman regarding the representation of functions whose ...

Shape-Measure Method for Solving Elliptic Optimal Shape Problems (Fixed Control Case)

Renormalized Solutions for Strongly Nonlinear Elliptic Problems with Lower Order Terms and Measure Data in Orlicz-Sobolev Spaces

The purpose of this paper is to prove the existence of a renormalized solution of perturbed elliptic problems$ -operatorname{div}Big(a(x,u,nabla u)+Phi(u) Big)+ g(x,u,nabla u) = mumbox{ in }Omega,  $ in the framework of Orlicz-Sobolev spaces without any restriction on the $M$ N-function of the Orlicz spaces, where $-operatorname{div}Big(a(x,u,nabla u)Big)$ is a Leray-Lions operator defined f...

Stanford Department of Mathematics Analysis & PDE seminar Dirichlet problem for elliptic operators with rough coefficients and L-harmonic measure

Sharp estimates for the solutions to elliptic PDEs in L∞ in terms of the corresponding norm of the boundary data follow directly from the maximum principle. It holds on arbitrary domains for all (real) second order divergence form elliptic operators −divA∇. The wellposedness of boundary problems in L, p < ∞, is a far more intricate and challenging question, even in a half-space. In particular, ...

A new approach to interior regularity of elliptic systems with quadratic Jacobian structure in dimension two

then u is continuous. This result solves a conjecture of Heinz about regularity of solutions to the prescribed boundedmean curvature equation and a conjecture of Hildebrandt about regularity of all critical points of continuously differentiable elliptic conformally invariant Lagrangians in dimension two. In particular it provides a new proof of Hélein’s theorem [10,12] about regularity of two d...