Georg S. Weiss

Partial regularity for a minimum problem with free boundary

AbstractRegularity of the free boundary ∂{u > 0} of a non-negative minimum u of the functional % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa8xXdi% ablAAiHnaapefabaWaaeWaaeaadaabdaqaaiabgEGirlaa-v8aaiaa% wEa7caGLiWoadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWFrbWaaW% baaSqabeaaieaacaGFYaaaaOGaa43XdmaaBaaaleaadaGadaqaaiaa% -v8acaWF+aGaa4hmaaGaay5Eaiaaw2haaaqabaaakiaawIcacaGLPa% aaaSqaaiabfM6axbqab0Gaey4kIipaaaa!4E4E!

Self-similar blow-up and Hausdorff dimension estimates for a class of parabolic free boundary problems

\upsilon \mapsto \int\limits_\Omega {\left( {\left| {\nabla \upsilon } \right|^2 + Q^2 \chi _{\left\{ {\upsilon > 0} \right\}} } \right)}

Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem

, where Ω is an open set in ℝn and Q is a strictly positive Hölder-continuous function, is still an open problem for n ≥ 3.By means of a new monotonicity formula we prove that the existence of singularities is equivalent to the existence of an absolute minimum u* such that the graph of u* is a cone with vertex at 0, the free boundary ∂{u* > 0} has one and only one singularity, and the set {u* > 0} minimizes the perimeter among all its subsets.This leads to the following partial regularity: there is a maximal dimension k* ≥ 3 such that for n < k* the free boundary ∂{u > 0} is locally in Ω a C1,α-surface, for n = k* the singular set Σ:= ∂{u > 0} − ∂red{u > 0} consists at most of in Ω isolated points, and for n > k* the Hausdorff dimension of the singular set Σ is less than n - k*.

A Parabolic Free Boundary Problem with Bernoulli Type Condition on the Free Boundary

where λ+ 0 and λ+ + λ− > 0. Equation (1) is related to the time-dependent equation 0 = α∂t max(v, 0)+ β∂t min(v, 0)−∆v in (0, T ) × Ω which has been used to describe an instantaneous and complete reaction of two substances coming into contact at a surface Γ (see [2, 3] and [6]). The difficulty one confronts in this Stefan-like problem is that the interface {v = 0} consists in general of two parts—one where the gradient of v is nonzero and one where the gradient of v vanishes. At the latter part we expect the gradient of v to have linear growth in space. However, because of the decomposition into two different types of growth, it is not possible to derive a growth estimate by, for example, a Bernstein technique. Assuming that α > β > 0 and that the time derivative ∂tv is non-negative and Holder continuous near some free boundary point (t0, x0), v(t0) is a solution of (1) with Holder continuous coefficients λ+ and λ−. To that our result applies and yields the expected C1,1-regularity of v(t0) in a pointwise sense and, for positive time derivative, the Hausdorff dimension estimate dim(∂{v(t0) > 0} ∪ ∂{v(t0) < 0}) n − 1 (see Proposition 4.1, Remark 4.1, Corollary 5.1 and Remark 5.1).

Double obstacle problems with obstacles given by non-C 2 Hamilton–Jacobi equations

We consider variational solutions of the quenching problem

A remark on the geometry of uniformly rotating stars

\partial_t u - \Delta u = - u^\gamma \chi_{\{u>0\}}

THE SINGULAR SET OF HIGHER DIMENSIONAL UNSTABLE OBSTACLE TYPE PROBLEMS

with exponent

Aleksandrov and Kelvin Reflection and the Regularity of Free Boundaries

\gamma \in (-{1\over 3},0)