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Register a new userInhomogeneous global minimizers to the one-phase free boundary problem
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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.
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We consider a free boundary problem on three-dimensional cones depending on a parameter c and study when the free boundary is allowed to pass through the vertex of the cone. Combining analysis and computer-assisted proof, we show that when c is less than 0.43, the free boundary may pass through the vertex of the cone.
We consider almost minimizers to the thin-one phase energy functional and we prove optimal regularity of the solution and partial regularity of the free boundary. We thus recover the theory for energy minimizers. Our methods are based on a noninfinitesimal notion of viscosity solutions.
The free boundary problem for a two-dimensional fluid filtered in porous media is studied. This is known as the one-phase Muskat problem and is mathematically equivalent to the vertical Hele-Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global-in-time Lipschitz solution in the strong $L^infty_t L^2_x$ sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.
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.
We investigate the regularity of the free boundary for the Signorini problem in $mathbb{R}^{n+1}$. It is known that regular points are $(n-1)$-dimensional and $C^infty$. However, even for $C^infty$ obstacles $varphi$, the set of non-regular (or degenerate) points could be very large, e.g. with infinite $mathcal{H}^{n-1}$ measure. The only two assumptions under which a nice structure result for degenerate points has been established are: when $varphi$ is analytic, and when $Deltavarphi < 0$. However, even in these cases, the set of degenerate points is in general $(n-1)$-dimensional (as large as the set of regular points). In this work, we show for the first time that, usually, the set of degenerate points is small. Namely, we prove that, given any $C^infty$ obstacle, for almost every solution the non-regular part of the free boundary is at most $(n-2)$-dimensional. This is the first result in this direction for the Signorini problem. Furthermore, we prove analogous results for the obstacle problem for the fractional Laplacian $(-Delta)^s$, and for the parabolic Signorini problem. In the parabolic Signorini problem, our main result establishes that the non-regular part of the free boundary is $(n-1-alpha_circ)$-dimensional for almost all times $t$, for some $alpha_circ > 0$. Finally, we construct some new examples of free boundaries with degenerate points.
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