Related papers: Removable singularities for Lipschitz caloric func…
We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on…
In this work, we study the removability of boundary singular sets for certain classes of quasilinear elliptic equations in domains $\Omega$ of an $n$-dimensional Finsler manifold ( $\mathcal{M}, F, \vartheta$ ). We work with Lipschitz…
We introduce some new classes of time dependent functions whose defining properties take into account of oscillations around singularities. We study properties of solutions to the heat equation with coefficients in these classes which are…
We bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time. In particular we characterize those equations for which the Lipschitz…
In this article, the boundary singularity for stationary solutions of the linearized Boltzmann equation with cut-off inverse power potential is analyzed. In particular, for cut-off hard-potential cases, we establish the asymptotic…
We give matching upper and lower bounds for the Dirichlet heat kernel of a Schr\"odinger operator $\Delta+W$ in the domain above the graph of a bounded Lipschitz function, in the case when $W$ decays away from the boundary faster than…
We prove several functional and geometric inequalities only assuming the linearity and a quantitative $\mathrm{L}^\infty$-to-Lipschitz smoothing of the heat semigroup in metric-measure spaces. Our results comprise a Buser inequality, a…
We consider the Cauchy problem for strictly hyperbolic $m$-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that the problem is $L^2$ well-posed in the case of Lipschitz…
We study the heat capacity of a static system of self-gravitating radiations analytically in the context of general relativity. To avoid the complexity due to a conical singularity at the center, we excise the central part and replace it…
We study the nonhomogeneous Dirichlet problem for first order Hamilton-Jacobi equations associated with Tonelli Hamiltonians on a bounded domain $\Omega$ of $\R^n$ assuming the energy level to be supercritical. First, we show that the…
We use some results from the theory of Reproducing Kernel Hilbert Spaces to show that the reachable space of the heat equation for a finite rod with either one or two Dirichlet boundary controls is a RKHS of analytic functions on a square,…
We study inverse boundary problems for a one dimensional linear integro-differential equation of the Gurtin--Pipkin type with the Dirichlet-to-Neumann map as the inverse data. Under natural conditions on the kernel of the integral operator,…
In this paper we study singular integrals on small (that is, measure zero and lower than full dimensional) subsets of metric groups. The main examples of the groups we have in mind are Euclidean spaces and Heisenberg groups. In addition to…
Let $L$ be a closed, densely defined operator of type $ \omega $ on $ L^2(\mathbb{R}^n)$ with $0 \leq \omega < \pi/2 $. We assume that $ L $ possesses a bounded $ H_\infty $-functional calculus and that its heat kernel satisfies suitable…
In the present paper, a systematic study is made of quantitative semicontinuity (a.k.a. Lipschitzian) properties of certain multifunctions, which are defined as a solution map associated to a family of parameterized ``split" feasibility…
We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We…
We construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equations with variable coefficients on the entire real line. The corresponding Green function (heat kernel) is given in terms of elementary…
We study the correct solvability of an abstract functional differential equations in Hilbert space, which includes integro-differential equations describing evolution of thermal phenomena, heat transfer in materials with memory or sound…
We prove the necessary and sufficient condition for the removability of the fundamental singularity, and equivalently for the unique solvability of the singular Dirichlet problem for the heat equation. In the measure-theoretical context,…
In this work, we are interested in to study removability of a singular set in the boundary for some classes of quasilinear elliptic equations. We will approach this question in two different ways: through an asymptotic behavior at the…