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Register a new userExistence of parabolic minimizers to the total variation flow on metric measure spaces
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We give an existence proof for variational solutions $u$ associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $(mathcal{X}, d, mu)$ satisfying a doubling condition and supporting a Poincare inequality. For such parabolic minimizers that coincide with a time-independent Cauchy-Dirichlet datum $u_0$ on the parabolic boundary of a space-time-cylinder $Omega times (0, T)$ with $Omega subset mathcal{X}$ an open set and $T > 0$, we prove existence in the weak parabolic function space $L^1_w(0, T; mathrm{BV}(Omega))$. In this paper, we generalize results from a previous work by Bogelein, Duzaar and Marcellini by introducing a more abstract notion for $mathrm{BV}$-valued parabolic function spaces. We argue completely on a variational level.
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In this note we consider problems related to parabolic partial differential equations in geodesic metric measure spaces, that are equipped with a doubling measure and a Poincare inequality. We prove a location and scale invariant Harnack inequality for a minimizer of a variational problem related to a doubly non-linear parabolic equation involving the p-Laplacian. Moreover, we prove the sufficiency of the Grigoryan--Saloff-Coste theorem for general p > 1 in geodesic metric spaces. The approach used is strictly variational, and hence we are able to carry out the argument in the metric setting.
In this paper we study the Total Variation Flow (TVF) in metric random walk spaces, which unifies into a broad framework the TVF on locally finite weighted connected graphs, the TVF determined by finite Markov chains and some nonlocal evolution problems. Once the existence and uniqueness of solutions of the TVF has been proved, we study the asymptotic behaviour of those solutions and, with that aim in view, we establish some inequalities of Poincar{e} type. In particular, for finite weighted connected graphs, we show that the solutions reach the average of the initial data in finite time. Furthermore, we introduce the concepts of perimeter and mean curvature for subsets of a metric random walk space and we study the relation between isoperimetric inequalities and Sobolev inequalities. Moreover, we introduce the concepts of Cheeger and calibrable sets in metric random walk spaces and characterize calibrability by using the $1$-Laplacian operator. Finally, we study the eigenvalue problem whereby we give a method to solve the optimal Cheeger cut problem.
Let (X j , d j , $mu$ j), j = 0, 1,. .. , m be metric measure spaces. Given 0 < p $kappa$ $le$ $infty$ for $kappa$ = 1,. .. , m and an analytic family of multilinear operators T z : L p 1 (X 1) x $bullet$ $bullet$ $bullet$ L p m (X m) $rightarrow$ L 1 loc (X 0), for z in the complex unit strip, we prove a theorem in the spirit of Steins complex interpolation for analytic families. Analyticity and our admissibility condition are defined in the weak (integral) sense and relax the pointwise definitions given in [9]. Continuous functions with compact support are natural dense subspaces of Lebesgue spaces over metric measure spaces and we assume the operators T z are initially defined on them. Our main lemma concerns the approximation of continuous functions with compact support by similar functions that depend analytically in an auxiliary parameter z. An application of the main theorem concerning bilinear estimates for Schr{o}dinger operators on L p is included.
In this paper, we will prove the Weyls law for the asymptotic formula of Dirichlet eigenvalues on metric measure spaces with generalized Ricci curvature bounded from below.
This paper proves the strong parabolic Harnack inequality for local weak solutions to the heat equation associated with time-dependent (nonsymmetric) bilinear forms. The underlying metric measure Dirichlet space is assumed to satisfy the volume doubling condition, the strong Poincare inequality, and a cutoff Sobolev inequality. The metric is not required to be geodesic. Further results include a weighted Poincare inequality, as well as upper and lower bounds for non-symmetric heat kernels.
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