Yumi Cho, Christoph Scheven
We prove local higher integrability of the spatial gradient for solutions to obstacle problems of porous medium type in the fast diffusion case $m<1$. The result holds for the natural range of exponents that is known from other regularity results for porous medium type equations. We also cover the case of signed solutions.
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This paper is concerned with the self-improving property for obstacle problem related to the singular porous medium equation. We establish a local higher integrability result for the spatial gradient of the $m$-th power of nonnegative weak solutions, under some suitable regularity assumptions on the obstacle function.
We prove continuity up to the boundary of the minimizer of an obstacle problem and higher integrability of its gradient under generalized Orlicz growth. The result recovers similar results obtained in the special cases of polynomial growth, variable exponent growth and produces new results for Orlicz and double phase growth.
We prove global $W^{1,q}(Ω,\mathbb{R}^N)$-regularity for minimisers of $\mathscr{F}(u)=\int_ΩF(x,\mathrm{D}u)\mathrm{d} x$ satisfying $u\geq ψ$ for a given Sobolev obstacle $ψ$. $W^{1,q}(Ω,\mathbb{R}^m)$ regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on $F(x,z)$ are a uniform $α$-Hölder continuity assumption in $x$ and natural $(p,q)$-growth conditions in $z$ with $q<\frac{(n+α)p}{n}$. In the autonomous case $F\equiv F(z)$ we can improve the gap to $q<\frac{np}{n-1}$, a result new even in the unconstrained case.
We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and Hölder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent $p(x)$ is Hölder continuous.
In the paper, we consider the obstacle problem, with one and two irregular barriers, for semilinear evolution equation involving measure data and operator corresponding to a semi-Dirichlet form. We prove the existence and uniqueness of solutions under the assumption that the right-hand side of the equation is monotone and satisfies mild integrability conditions. To treat the case of irregular barriers, we extend the theory of precise versions of functions introduced by M. Pierre. We also give some applications to the so-called switching problem.
In 2012, F.Leonetti and F.Siepe [1] considered solutions to boundary value problems of some anisotropic elliptic equations of the type $$ \left\{ \begin{array}{llll} \sum\limits _{i=1}\limits^{n} D_i (a_i(x,Du(x)))=0, &x\in Ω,\\ u(x)=θ(x), & x\in \partial Ω. \end{array} \right. $$ Under some suitable conditions, they obtained an integrability result, which shows that, higher integrability of the boundary datum $θ$ forces solutions $u$ to have higher integrability as well. In the present paper, we consider ${\cal K}_{ψ,θ}^{(p_i)}$-obstacle problems of the nonhomogeneous anisotropic elliptic equ…
Higher integrability for the singular porous medium system
本文在快扩散范围内，通过内禀尺度、反向Hölder不等式和Vitali覆盖论证，建立了多孔介质系统弱解空间梯度的高阶可积性，并给出显式反向Hölder不等式，适用于一般快扩散系统及非负与带符号解。
Dispersive estimates for wave and Schrödinger equations with a potential in non-trapping exterior domains
本文研究非捕获外部区域中带短程位势的波动方程与薛定谔方程，通过证明预解估计，得到局部能量可积性、光滑效应，并将自由方程的局部 Strichartz 估计推广为全局估计。
This paper examines the decay properties of positive solutions for a family of fully nonlinear systems of integral equations containing Wolf potentials and Hardy weights. This class of systems includes examples which are closely related to the Euler-Lagrange equations for several classical inequalities such as the Hardy-Sobolev and Hardy-Littlewood-Sobolev inequalities. In particular, a complete characterization of the fast decaying ground states in terms of their integrability is provided in that bounded and fast decaying solutions are shown to be equivalent to the integrable solutions. In ge…
Existence of weak solutions for fast diffusion equation with a divergence type of drift term
在漂移项满足可积性条件下，构造满足L^q能量不等式和Wasserstein空间速度估计的非负弱解；当漂移项无散时，可放宽条件，并应用于粘性Boussinesq系统。
Harnack-type estimates for nonlocal double phase functionals
针对非局部双相位泛函的局部极小化子，建立了Harnack型不等式，通过同心模估计解决了(p,q)-增长与低可积性指数的匹配问题，首次在自然结构条件下给出完整理论。
Consider the problem of inverse scattering of time-harmonic point sources from an infinite, penetrable rough interface with bounded obstacles buried in the lower half-space, where the interface is assumed to be a local perturbation of a planar surface. A novel version of the sampling method is proposed to simultaneously reconstruct the local perturbation of the rough interface and buried obstacles by constructing a modified near-field equation associated with a special rough surface, yielding a fast imaging algorithm. Numerical examples are presented to illustrate the effectiveness of the inve…
This paper is devoted to the study of the generalized impedance boundary conditions (GIBCs) for a strongly absorbing obstacle in the {\bf time} regime in two and three dimensions. The GIBCs in the time domain are heuristically derived from the corresponding conditions in the time harmonic regime. The latters are frequency dependent except the one of order 0; hence the formers are non-local in time in general. The error estimates in the time regime can be derived from the ones in the time harmonic regime when the frequency dependence is well-controlled. This idea is originally due to Nguyen and…
Global higher integrability for systems with $p$-growth structure in noncylindrical domains
本文研究了非柱状区域中具有 $p$-增长结构系统的 Cauchy-Dirichlet 问题，在满足特定正则性及几何条件下证明了 $Du$ 的全局高阶可积性，该结果在 $p=2$ 时亦属新发现。
This article introduces a novel "bootstrap domain-of-dependence" concept, according to which, for all time following a given illumination period of arbitrary duration, the wave field scattered by an obstacle is encoded in the history of boundary scattering events for a time-length equal to the diameter of the obstacle, measured in time units. Resulting solution bounds provide estimates on the solution values in terms of a short-time history record, and they establish super-algebraically fast decay (i.e., decay faster than any negative power of time) for a wide range of scattering obstacle--inc…
Geometric Estimates for Solutions of Semilinear Equations with Singular Potentials
本文研究了半线性方程中奇异势的解的几何估计，扩展了尖锐正则性和非退化估计到无界、变号源的情况，分析了非线性与源的最小积分性假设的相互作用。
Building upon the recent results in \cite{FoSp17} we provide a thorough description of the free boundary for the fractional obstacle problem in $\mathbb{R}^{n+1}$ with obstacle function $φ$ (suitably smooth and decaying fast at infinity) up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, if $φ$ is analytic, the problem reduces to the zero obstacle case dealt with in \cite{FoSp17} and therefore we retrieve the same results: (i) local finiteness of the $(n-1)$-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure), (ii) $\mathcal{H}^{n-1}$-rectifiabili…
We consider an inverse acoustic scattering problem in simultaneously recovering an embedded obstacle and its surrounding inhomogeneous medium by formally determined far-field data. It is shown that the knowledge of the scattering amplitude with a fixed incident direction and all observation angles along with frequencies from an open interval can be used to uniquely identify the embedded obstacle, sound-soft or sound-hard disregarding the surrounding medium. Furthermore, if the surrounding inhomogeneous medium is from an admissible class (still general), then the medium can be recovered as well…
A horizontal $N$-dimensional plane, having a diffusion of its own, exchanges with the lower half space. There, a reaction-diffusion process, modelled by a free boundary problem, takes place. We wish to understand whether, and how, the free boundary meets the plane. The origin of this problem is a two-dimensional reaction diffusion model proposed some time ago by the second author, in collaboration with H. Berestycki and L. Rossi, to model how biological invasions can be enhanced by a line of fast diffusion. Some counter-intuitive numerical simulations of this model, due to A.-C. Coulon, have b…
This article carries out a qualitative analysis on a system of integral equations of the Hardy--Sobolev type. Namely, results concerning Liouville type properties and the fast and slow decay rates of positive solutions for the system are established. For a bounded and decaying positive solution, it is shown that it either decays with the slow rates or the fast rates depending on its integrability. Particularly, a criterion for distinguishing integrable solutions from other bounded and decaying solutions in terms of their asymptotic behavior is provided. Moreover, related results on the optimal…
The cut-off resolvent can grow arbitrarily fast in obstacle scattering
研究探讨了在无光滑性假设下，截断余子式的增长速率可任意加快，通过构造特定障碍物证明了在不同波数下余子式范数的快速增长。
We show that the gradient of the $m$-power of a solution to a singular parabolic equation of porous medium-type (also known as fast diffusion equation), satisfies a reverse Hölder inequality in suitable intrinsic cylinders. Relying on an intrinsic Calderón-Zygmund covering argument, we are able to prove the local higher integrability of such a gradient for $m\in\left(\frac{(n-2)_+}{n+2},1\right)$. Our estimates are satisfied for a general class of growth assumptions on the non linearity. In this way, we extend the theory for $m\geq 1$ (see [GS16] in the list of references) to the singular case…
We prove an existence and uniqueness result for solutions to nonlinear diffusion equations with degenerate mobility posed on a bounded interval for a certain density $u$. In case of \emph{fast-decay} mobilities, namely mobilities functions under a Osgood integrability condition, a suitable coordinate transformation is introduced and a new nonlinear diffusion equation with linear mobility is obtained. We observe that the coordinate transformation induces a mass-preserving scaling on the density and the nonlinearity, described by the original nonlinear mobility, is included in the diffusive proc…
This paper is concerned with reconstructing an acoustic obstacle and its excitation sources from the phaseless near-field measurements. By supplementing some artificial sources to the inverse scattering system, this co-inversion problem can be decoupled into two inverse problems: an inverse obstacle scattering problem and an inverse source problem, and the corresponding uniqueness can be established. This novel decoupling technique requires some extra data but brings in several salient benefits. First, our method is fast and easy to implement. Second, the boundary condition of the obstacle is …
We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for $d \geq 2$ for the Laplace operator $Δ$ with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators $Δ_1$ and $Δ_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative trace operator $g(Δ) - g(Δ_1) - g(Δ_2) + g(Δ_0)$ was introduced in [18] and shown to be trace-class for a large class of functions $g$, including certrain functions of polynomial growth. When $g$ is sufficiently regular at zero and fast decaying at infi…
We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for $d \geq 2$. In this setting the absolutely continuous part of the Laplace operator $Δ$ with Dirichlet boundary conditions and the free Laplace operator $Δ_0$ are unitarily equivalent. For suitable functions that decay sufficiently fast we have that the difference $g(Δ)-g(Δ_0)$ is a trace-class operator and its trace is described by the Krein spectral shift function. In this paper we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative…
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