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Register a new userParabolic integrodifferential identification problems related to radial memory kernels I
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We are concerned with the problem of recovering the radial kernel $k$, depending also on time, in a parabolic integro-differential equation $$D_{t}u(t,x)={cal A}u(t,x)+int_0^t k(t-s,|x|){cal B}u(s,x)ds +int_0^t D_{|x|}k(t-s,|x|){cal C}u(s,x)ds+f(t,x),$$ ${cal A}$ being a uniformly elliptic second-order linear operator in divergence form. We single out a special class of operators ${cal A}$ and two pieces of suitable additional information for which the problem of identifying $k$ can be uniquely solved locally in time when the domain under consideration is a spherical corona or an annulus.
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We are concerned with the problem of recovering the radial kernel $k$, depending also on time, in the parabolic integro-differential equation $$D_{t}u(t,x)={cal A}u(t,x)+int_0^t k(t-s,|x|){cal B}u(s,x)ds +int_0^t D_{|x|}k(t-s,|x|){cal C}u(s,x)ds+f(t,x),$$ ${cal A}$ being a uniformly elliptic second-order linear operator in divergence form. We single out a special class of operators ${cal A}$ and two pieces of suitable additional information for which the problem of identifying $k$ can be uniquely solved locally in time when the domain under consideration is a ball or a disk.
The tangential condition was introduced in [Hanke et al., 95] as a sufficient condition for convergence of the Landweber iteration for solving ill-posed problems. In this paper we present a series of time dependent benchmark inverse problems for which we can verify this condition.
We study the obstacle problem for parabolic operators of the type $partial_t + L$, where $L$ is an elliptic integro-differential operator of order $2s$, such as $(-Delta)^s$, in the supercritical regime $s in (0,frac{1}{2})$. The best result in this context was due to Caffarelli and Figalli, who established the $C^{1,s}_x$ regularity of solutions for the case $L = (-Delta)^s$, the same regularity as in the elliptic setting. Here we prove for the first time that solutions are actually textit{more} regular than in the elliptic case. More precisely, we show that they are $C^{1,1}$ in space and time, and that this is optimal. We also deduce the $C^{1,alpha}$ regularity of the free boundary. Moreover, at all free boundary points $(x_0,t_0)$, we establish the following expansion: $$(u - varphi)(x_0+x,t_0+t) = c_0(t - acdot x)_+^2 + O(t^{2+alpha}+|x|^{2+alpha}),$$ with $c_0 > 0$, $alpha > 0$ and $a in mathbb R^n$.
In this paper we consider the homogenization of the evolution problem associated with a jump process that involves three different smooth kernels that govern the jumps to/from different parts of the domain. We assume that the spacial domain is divided into a sequence of two subdomains $A_n cup B_n$ and we have three different smooth kernels, one that controls the jumps from $A_n$ to $A_n$, a second one that controls the jumps from $B_n$ to $B_n$ and the third one that governs the interactions between $A_n$ and $B_n$.Assuming that $chi_{A_n} (x) to X(x)$ weakly in $L^infty$ (and then $chi_{B_n} (x) to 1-X(x)$ weakly in $L^infty$) as $n to infty$ and that the initial condition is given by a density $u_0$ in $L^2$ we show that there is an homogenized limit system in which the three kernels and the limit function $X$ appear. When the initial condition is a delta at one point, $delta_{bar{x}}$ (this corresponds to the process that starts at $bar{x}$) we show that there is convergence along subsequences such that $bar{x} in A_{n_j}$ or $bar{x} in B_{n_j}$ for every $n_j$ large enough. We also provide a probabilistic interpretation of this evolution equation in terms of a stochastic process that describes the movement of a particle that jumps in $Omega$ according to the three different kernels and show that the underlying process converges in distribution to a limit process associated with the limit equation. We focus our analysis in Neumann type boundary conditions and briefly describe at the end how to deal with Dirichlet boundary conditions.
In this paper we consider second order parabolic partial differential equations subject to the Dirichlet boundary condition on smooth domains. We establish weighted $L_{q}$-maximal regularity in weighted Triebel-Lizorkin spaces for such parabolic problems with inhomogeneous boundary data. The weights that we consider are power weights in time and space, and yield flexibility in the optimal regularity of the initial-boundary data, allow to avoid compatibility conditions at the boundary and provide a smoothing effect. In particular, we can treat rough inhomogeneous boundary data.
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