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Register a new userHalf-space solutions with $7/2$ frequency in the thin obstacle problem
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For the thin obstacle problem in 3d, we show that half-space solutions form an isolated family in the space of $7/2$-homogeneous solutions. For a general solution with one blow-up profile in this family, we establish the rate of convergence to this profile. As a consequence, we obtain regularity of the free boundary near such contact points.
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In this paper we give a full classification of global solutions of the obstacle problem for the fractional Laplacian (including the thin obstacle problem) with compact coincidence set and at most polynomial growth in dimension $N geq 3$. We do this in terms of a bijection onto a set of polynomials describing the asymptotics of the solution. Furthermore we prove that coincidence sets of global solutions that are compact are also convex if the solution has at most quadratic growth.
We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation [ (-Delta)^s u = chi_{{u>c}} ] for $0<s<1$ and an arbitrary constant $c$. Our main object of study is the free boundary $partial{u>c}$. We study the behavior of the free boundary and prove an upper bound for the Hausdorff dimension of the singular set. We also show that when $sleq 1/2$ certain symmetric solutions are stable; however, when $s>1/2$ these solutions are not stable and therefore not minimizers of the corresponding functional.
The objective of this paper is twofold. First we provide the -- to the best knowledge of the authors -- first result on the behavior of the regular part of the free boundary of the obstacle problem close to singularities. We do this using our second result which is the partial answer to a long standing conjecture and the first partial classification of global solutions of the obstacle problem with unbounded coincidence sets.
We study a nonlinear equation in the half-space ${x_1>0}$ with a Hardy potential, specifically [-Delta u -frac{mu}{x_1^2}u+u^p=0quadtext{in}quad mathbb R^n_+,] where $p>1$ and $-infty<mu<1/4$. The admissible boundary behavior of the positive solutions is either $O(x_1^{-2/(p-1)})$ as $x_1to 0$, or is determined by the solutions of the linear problem $-Delta h -frac{mu}{x_1^2}h=0$. In the first part we study in full detail the separable solutions of the linear equations for the whole range of $mu$. In the second part, by means of sub and supersolutions we construct separable solutions of the nonlinear problem which behave like $O(x_1^{-2/(p-1)})$ near the origin and which, away from the origin have exactly the same asymptotic behavior as the separable solutions of the linear problem. In the last part we construct solutions that behave like $O(x_1^{-2/(p-1)})$ at some prescribed parts of the boundary, while at the rest of the boundary the solutions decay or blowup at a slower rate determined by the linear part of the equation.
The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the assets prices are driven by pure jump Levy processes. In this paper we study the regularity of the free boundary. Our main result establishes that, when $s>frac12$, the free boundary is a $C^{1,alpha}$ graph in $x$ and $t$ near any regular free boundary point $(x_0,t_0)in partial{u>varphi}$. Furthermore, we also prove that solutions $u$ are $C^{1+s}$ in $x$ and $t$ near such points, with a precise expansion of the form [u(x,t)-varphi(x)=c_0bigl((x-x_0)cdot e+a(t-t_0)bigr)_+^{1+s}+obigl(|x-x_0|^{1+s+alpha}+ |t-t_0|^{1+s+alpha}bigr),] with $c_0>0$, $ein mathbb{S}^{n-1}$, and $a>0$.
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