arXiv subjects

Wolfgang Arendt

Publications and source records attributed to Wolfgang Arendt.

27 records · Page 2Linked to original sources

arXiv2016

Mapping theorems for Sobolev-spaces of vector-valued functions

We consider Sobolev spaces with values in Banach spaces as they are frequently useful in applied problems. Given two Banach spaces $X\neq\{0\}$ and $Y$, each Lipschitz continuous mapping $F:X\rightarrow Y$ gives rise to a mapping $u\mapsto F\circ u$ from $W^{1,p}(Ω,X)$ to $W^{1,p}(Ω,Y)$ if and only if $Y$ has the Radon-Nikodym Property. But if $F$ is one-sided Gateaux differentiable no condition on the space is needed. We also study when weak properties in the sense of duality imply strong properties. Our results are applied to prove embedding theorems, a multi-dimensional version of the Aubin-Lions Lemma and characterizations of the space $W^{1,p}_0(Ω,X)$.

math.FA↗

arXiv2016

Diffusion with nonlocal boundary conditions

We consider second order differential operators $A_μ$ on a bounded, Dirichlet regular set $Ω\subset \mathbb{R}^d$, subject to the nonlocal boundary conditions \[ u(z) = \int_Ωu(x)\, μ(z, dx)\quad \mbox{for } z \in \partial Ω. \] Here the function $μ: \partialΩ\to \mathscr{M}^+(Ω)$ is $σ(\mathscr{M} (Ω), C_b(Ω))$-continuous with $0\leq μ(z,Ω) \leq 1$ for all $z\in \partial Ω$. Under suitable assumptions on the coefficients in $A_μ$, we prove that $A_μ$ generates a holomorphic positive contraction semigroup $T_μ$ on $L^\infty(Ω)$. The semigroup $T_μ$ is never strongly continuous, but it enjoys the strong Feller property in the sense that it consists of kernel operators and takes values in $C(\barΩ)$. We also prove that $T_μ$ is immediately compact and study the asymptotic behavior of $T_μ(t)$ as $t \to \infty$.

math.FA↗

arXiv2015

Maximal regularity for non-autonomous Robin boundary conditions

We consider a non-autonomous Cauchy problem involving linear operators associated with time-dependent forms $a(t;.,.):V\times V\to {\mathbb{C}}$ where $V$ and $H$ are Hilbert spaces such that $V$ is continuously embedded in $H$. We prove $H$-maximal regularity under a new regularity condition on the form $a$ with respect to time; namely H{ö}lder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.

math.AP↗

arXiv2014

Maximal Regularity for Evolution Equations Governed by Non-Autonomous Forms

\begin{abstract}\label{abstract} We consider a non-autonomous evolutionary problem \[ \dot{u} (t)+\A(t)u(t)=f(t), \quad u(0)=u_0 \] where the operator $\A(t):V\to V^\prime$ is associated with a form $\fra(t,.,.):V\times V \to \R$ and $u_0\in V$. Our main concern is to prove well-posedness with maximal regularity which means the following. Given a Hilbert space $H$ such that $V$ is continuously and densely embedded into $H$ and given $f\in L^2(0,T;H)$ we are interested in solutions $u \in H^1(0,T;H)\cap L^2(0,T;V)$. We do prove well-posedness in this sense whenever the form is piecewise Lipschitz-continuous and symmetric. Moreover, we show that each solution is in $C([0,T];V)$. We apply the results to non-autonomous Robin-boundary conditions and also use maximal regularity to solve a quasilinear problem.

math.AP↗

arXiv2013

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.

math.AP↗

arXiv2013

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

We consider a non-autonomous form $\fra:[0,T]\times V\times V \to \C$ where $V$ is a Hilbert space which is densely and continuously embedded in another Hilbert space $H$. Denote by $\A(t) \in Ł(V,V')$ the associated operator. Given $f \in L^2(0,T, V')$, one knows that for each $u_0 \in H$ there is a unique solution $u\in H^1(0,T;V')\cap L^2(0,T;V)$ of $$\dot u(t) + \A(t) u(t) = f(t), \, \, u(0) = u_0.$$ %\begin{align*} %&\dot u(t) + \A(t)u(t)= f(t)\ %& u(0)=u_0. %\end{align*} 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 $\Conv$ of $H$; i.e.\ we give a criterion on the form which implies that $u(t)\in \Conv$ for all $t\in[0,T]$ whenever $u_0\in\Conv$. In the autonomous case for $f = 0$, the criterion is known and even equivalent to invariance by a result proved in \cite{Ouh96} (see also \cite{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.

math.AP↗

arXiv2011

From forms to semigroups

Form methods give a very efficient tool to solve evolutionary problems on Hilbert space. They were developed by T. Kato [Kat] and, in slightly different language by J.L. Lions. In this expository article we give an introduction based on [AE2]. The main point in our approach is that the notion of closability is not needed anymore. The new setting is particularly efficient for the Dirichlet-to-Neumann operator and degenerate equations. Besides this we give several other examples. This presentation starts by an introduction to holomorphic semigroups. Instead of the contour argument found in the literature, we give a more direct argument based on the Hille--Yosida theorem.

math.AP↗

arXiv2010

Semigroups Generated by Elliptic Operators in Non-Divergence Form on $C_0(/omega)$

Given a bounded domain in the Euclidean space satisfying the uniform outer cone condition, we show that a uniformly elliptic operator of second order with continuous second order coefficients generates a holomorphic semigroup on the space of all continuous functions vanishing at the boundary. In particular, Lipschitz domains are allowed. This result was so far known under considerable stronger regularity assumptions. Also the Dirichlet problem is considered for such operators.

math.AP↗

arXiv2008

Diffusion determines the manifold

We prove under a weak smoothness condition that two Riemannian manifold are isomorphic if and only there exists an order isomorphism which intertwines with the Dirichlet type heat semigroups on the manifolds.

math.AP↗