Regularity of Lipschitz free boundaries for the thin one-phase problem
We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional among all functions which are fixed on .
We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional among all functions which are fixed on .
We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of in a smooth bounded domain . In particular, we obtain new and bounds for the extremal solution when the domain is strictly convex. More precisely, we prove that if and if .
It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding configurations. We then use this to construct weak solutions to the unstable interface problem (the...
Classical Gårding inequalities such as those of Hörmander, Hörmander-Melin or Fefferman-Phong are proved by pseudo-differential methods which do not allow to keep a good control on the supports of the functions under study nor on the smoothness of the coefficients of the operator. In this paper, we show by very simple calculations that in certain special situations, the results that can be obtained directly are much better than those expected thanks to the general theory.
We consider Schrödinger operators on with variable coefficients. Let be the free Schrödinger operator and we suppose is a “short-range” perturbation of . Then, under the nontrapping condition, we show that the time evolution operator: can be written as a product of the free evolution operator and a Fourier integral operator which is associated to the canonical relation given by the classical mechanical scattering. We also prove a similar result for the wave operators. These results...
In compressible Neohookean elasticity one minimizes functionals which are composed by the sum of the norm of the deformation gradient and a nonlinear function of the determinant of the gradient. Non–interpenetrability of matter is then represented by additional invertibility conditions. An existence theory which includes a precise notion of invertibility and allows for cavitation was formulated by Müller and Spector in 1995. It applies, however, only if some -norm of the gradient with is controlled...
With help of suitable anisotropic Minkowski’s contents and Hausdorff measures some results are obtained concerning removability of singularities for solutions of partial differential equations with anisotropic growth in the vicinity of the singular set.
1. Introduction. The study of singularities has been one of the main subjects of research in partial differential equations. In the case of linear equations the singularities are now pretty well understood; but in the nonlinear case there seems to be still very few studies. In this paper I want to discuss the singularities of solutions of a class of nonlinear singular partial differential equations in the complex domain. The class is only a model, but it helps one understand that the situation in...