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Register a new userOn certain degenerate one-phase free boundary problems
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We develop an existence and regularity theory for a class of degenerate one-phase free boundary problems. In this way we unify the basic theories in free boundary problems like the classical one-phase problem, the obstacle problem, or more generally for minimizers of the Alt-Phillips functional.
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Given a global 1-homogeneous minimizer $U_0$ to the Alt-Caffarelli energy functional, with $sing(F(U_0)) = {0}$, we provide a foliation of the half-space $R^{n} times [0,+infty)$ with dilations of graphs of global minimizers $underline U leq U_0 leq bar U$ with analytic free boundaries at distance 1 from the origin.
For a vectorial Bernoulli-type free boundary problem, with no sign assumption on the components, we prove that flatness of the free boundary implies $C^{1,alpha}$ regularity, as well-known in the scalar case cite{AC,C2}. While in cite{MTV2} the same result is obtained for minimizing solutions by using a reduction to the scalar problem, and the NTA structure of the regular part of the free boundary, our result uses directly a viscosity approach on the vectorial problem, in the spirit of cite{D}. We plan to use the approach developed here in vectorial free boundary problems involving a fractional Laplacian, as those treated in the scalar case in cite{DR, DSS}.
We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [4,8]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form $J_epsilon(x)=frac 1epsilon J(frac xepsilon)$ for small $epsilon>0$, where $J(x)$ has compact support. We also give an estimate of the error term of the approximation by some positive power of $epsilon$.
This is an introduction to The Theme Issue on Free Boundary Problems and Related Topics, which consists of 14 survey/review articles on the topics, of Philosophical Transactions of the Royal Society A: Physical, Mathematical and Engineering Sciences, 373, no. 2050, The Royal Society, 2015.
The semi-geostrophic system is widely used in the modelling of large-scale atmospheric flows. In this paper, we prove existence of solutions of the incompressible semi-geostrophic equations in a fully three-dimensional domain with a free upper boundary condition. We show that, using methods similar to those introduced in the pioneering work of Benamou and Brenier, who analysed the same system but with a rigid boundary condition, we can prove the existence of solutions for the incompressible free boundary problem. The proof is based on optimal transport results as well as the analysis of Hamiltonian ODEs in spaces of probability measures given by Ambrosio and Gangbo. We also show how these techniques can be modified to yield the same result also for the compressible version of the system.
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