On the spectrum of the -biharmonic operator.
On the stability of the free boundary in the obstacle problem for a minimal surface
On the structure of the conformal Gaussian curvature equation on R2. II.
On the symmetry of blowup solutions to a mean field equation
On the unique solvability of nonresonant elliptic equations
On the uniqueness of solutions of the homogeneous curvature equations
On the uniqueness problem for quasilinear elliptic equations involving measures.
We discuss the uniqueness of solutions to problems like⎧ λ |u|s-1u - div (|∇u|p-2∇u) = μ on Ω,⎨⎩ u = 0 in ∂Ω,where λ ≥ 0 and μ is a signed Radon measure.
On the -regularity of solutions to systems of differential equations in the case when the equations are constructed from discontinuous functions.
On two problems studied by A. Ambrosetti
We study the Ambrosetti–Prodi and Ambrosetti–Rabinowitz problems.We prove for the first one the existence of a continuum of solutions with shape of a reflected (-shape). Next, we show that there is a relationship between these two problems.
On Two-Dimensional Elliptic Systems with a One-sided Condition.
On Uniqueness of Positive Solutions of Nonlinear Elliptic Boundary Value Problems.
On Univalent Harmonic Maps Between Surfaces.
On weakly A-harmonic tensors
We study very weak solutions of an A-harmonic equation to show that they are in fact the usual solutions.
One dimensional symmetry in the Heisenberg group
One-dimensional symmetry for solutions of quasilinear equations in
In this paper we consider two-dimensional quasilinear equations of the form and study the properties of the solutions u with bounded and non-vanishing gradient. Under a weak assumption involving the growth of the argument of (notice that is a well-defined real function since on ) we prove that is one-dimensional, i.e., for some unit vector . As a consequence of our result we obtain that any solution having one positive derivative is one-dimensional. This result provides a proof of...
One-dimensional symmetry of periodic minimizers for a mean field equation
We consider on a two-dimensional flat torus defined by a rectangular periodic cell the following equationIt is well-known that the associated energy functional admits a minimizer for each . The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever . Our results hold more generally for solutions that are Steiner symmetric, up to a translation....
Operator preconditioning with efficient applications for nonlinear elliptic problems
This paper is devoted to the numerical solution of nonlinear elliptic partial differential equations. Such problems describe various phenomena in science. An approach that exploits Hilbert space theory in the numerical study of elliptic PDEs is the idea of preconditioning operators. In this survey paper we briefly summarize the main lines of this theory with various applications.
Optimal control of nonsmooth system governed by quasi-linear elliptic equations.
Optimal interior partial regularity for nonlinear elliptic systems for the case under natural growth condition.
Optimal regularity for the pseudo infinity Laplacian
In this paper we find the optimal regularity for viscosity solutions of the pseudo infinity Laplacian. We prove that the solutions are locally Lipschitz and show an example that proves that this result is optimal. We also show existence and uniqueness for the Dirichlet problem.