Non-Autonomous Maximal Regularity for Forms of Bounded Variation

We consider a non-autonomous evolutionary problem $$u' (t)+\mathcal A (t)u(t)=f(t), \quad u(0)=u_0,$$ where $V, H$ are Hilbert spaces such that $V$ is continuously and densely embedded in $H$ and the operator $\mathcal A (t)\colon V\to V^\prime$ is associated with a coercive, bounded, symmetric form $\mathfrak{a}(t,.,.)\colon V\times V \to \mathbb{C}$ for all $t \in [0,T]$. Given $f \in L^2(0,T;H)$, $u_0\in V$ there exists always a unique solution $u \in MR(V,V'):= L^2(0,T;V) \cap H^1(0,T;V')$. The purpose of this article is to investigate when $u \in H^1(0,T;H)$. This property of maximal regularity in $H$ is not known in general. We give a positive answer if the form is of bounded variation; i.e., if there exists a bounded and non-decreasing function $g \colon [0,T] \to \mathbb{R}$ such that \begin{equation*} \lvert\mathfrak{a}(t,u,v)- \mathfrak{a}(s,u,v)\rvert \le [g(t)-g(s)] \lVert u \rVert_V \lVert v \rVert_V \quad (s,t \in [0,T], s \le t). \end{equation*} In that case, we also show that $u(.)$ is continuous with values in $V$. Moreover we extend this result to certain perturbations of $\mathcal A (t)$.Comment: 22 page

Diffusion in networks with time-dependent transmission conditions

We study diffusion in a network which is governed by non-autonomous Kirchhoff conditions at the vertices of the graph. Also the diffusion coefficients may depend on time. We prove at first a result on existence and uniqueness using form methods. Our main results concern the long-term behavior of the solution. In the case when the conductivity and the diffusion coefficients match (so that mass is conserved) we show that the solution converges exponentially fast to an equilibrium. We also show convergence to a special solution in some other cases.Comment: corrected typos, references removed, revised Lemma A.3. Appl. Math. Optim. (2013

Non-autonomous Cauchy problems governed by forms: maximal regularity and invariance

Form methods are a useful and elegant framework to study second order elliptic operators in divergence form. They can be used to describe such operators including various boundary conditions, such as Dirichlet, Neumann and Robin boundary conditions. In the autonomous case Cauchy problems of the form u´(t) + Au(t) = f (t), u(0) = u_0, where A is associated with a form a, are well studied. The subject of this thesis are non-autonomous Cauchy problems associated with a form a(t) depending on t. We study regularity, invariance of convex sets and asymptotics

Discrete versions of the Li-Yau gradient estimate

Dier D, Kaßmann M, Zacher R. Discrete versions of the Li-Yau gradient estimate. arXiv:1701.04807. 2017.We study positive solutions to the heat equation on graphs. We prove variants of the Li-Yau gradient estimate and the differential Harnack inequality. For some graphs, we can show the estimates to be sharp. We establish new computation rules for differential operators on discrete spaces and introduce a relaxation function that governs the time dependency in the differential Harnack estimate

Invariance of Convex Sets for Non-autonomous Evolution Equations Governed by Forms

We consider a non-autonomous form a: [0, T] × V × V → C where V is a Hilbert space which is densely and continuously embedded in another Hilbert space H. Denote by A(t) ∈ L(V, V ′ ) the associated operator. Given f ∈ L 2 (0, T, V ′), one knows that for each u0 ∈ H there is a unique solution u ∈ H 1 (0, T; V ′ ) ∩ L 2 (0, T; V) of ˙u(t) + A(t)u(t) = f(t), u(0) = u0. This result by J. L. Lions is well-known. The aim of this article is to find a criterion for the invariance of a closed convex subset C of H; i.e. we give a criterion on the form which implies that u(t) ∈ C for all t ∈ [0, T] whenever u0 ∈ C. In the autonomous case for f = 0, the criterion is known and even equivalent to invariance by a result proved in [Ouh96] (see also [Ouh05]). We give applications to positivity and comparison of solutions to heat equations with non-autonomous Robin boundary conditions. We also prove positivity of the solution to a quasi-linear heat equation

Maximal Regularity for Evolution Equations Governed by Non-Autonomous Forms

We consider a non-autonomous evolutionary proble

On the parabolic Harnack inequality for non-local diffusion equations

Abstract We settle the open question concerning the Harnack inequality for globally positive solutions to non-local in time diffusion equations by constructing a counter-example for dimensions d ≥ β, where β ∈ (0,2] is the order of the equation with respect to the spatial variable. The equation can be non-local both in time and in space but for the counter-example it is important that the equation has a fractional time derivative. In this case, the fundamental solution is singular at the origin for all times t > 0 in dimensions d ≥ β. This underlines the markedly different behavior of time-fractional diffusion compared to the purely space-fractional case, where a local Harnack inequality is known. The key observation is that the memory strongly affects the estimates. In particular, if the initial data $\mathit{u}_0 \in \mathit{L}_{loc}^{q}$ for q larger than the critical value $\frac{d}{\beta}$ of the elliptic operator (−Δ)β/2, a non-local version of the Harnack inequality is still valid as we show. We also observe the critical dimension phenomenon already known from other contexts: the diffusion behavior is substantially different in higher dimensions than d = 1 provided β > 1, since we prove that the local Harnack inequality holds if d < β