Surfaces à courbure moyenne constante et inégalité de Wente
Symmetry and concentration behavior of ground state in axially symmetric domains.
Symmetry and convexity of level sets of solutions to the infinity Laplace's equation.
Symmetry and geometric evolution equations.
Symmetry and uniqueness of parabolic affine spheres.
Symmetry of minimizers with a level surface parallel to the boundary
We consider the functional where is a bounded domain and is a convex function. Under general assumptions on , Crasta [Cr1] has shown that if admits a minimizer in depending only on the distance from the boundary of , then must be a ball. With some restrictions on , we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these...
Symmetry of solutions of semilinear elliptic problems
We study symmetry properties of least energy positive or nodal solutions of semilinear elliptic problems with Dirichlet or Neumann boundary conditions. The proof is based on symmetrizations in the spirit of Bartsch, Weth and Willem (J. Anal. Math., 2005) together with a strong maximum principle for quasi-continuous functions of Ancona and an intermediate value property for such functions.
Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations
We study uniformly elliptic fully nonlinear equations , and prove results of Gidas–Ni–Nirenberg type for positive viscosity solutions of such equations. We show that symmetries of the equation and the domain are reflected by the solution, both in bounded and unbounded domains.
Symmetry theorems and uniform rectifiability.