Variational parabolic capacity (original) (raw)

Diffuse measures and nonlinear parabolic equations

Journal of Evolution Equations, 2011

Given a parabolic cylinder Q = (0, T ) × Ω, where Ω ⊂ R N is a bounded domain, we prove new properties of solutions of ut − ∆pu = µ in Q with Dirichlet boundary conditions, where µ is a finite Radon measure in Q.

Regularity for degenerate elliptic and parabolic systems

2013

In this work local behavior for solutions to the inhomogeneous p-Laplace in divergence form and its parabolic version are studied. It is parabolic and non-linear generalization of the Calderon-Zygmund theory for the Laplace operator. I.e. the borderline case BMO is studied. The two main results are local BMO and Hoelder estimates for the inhomogenious p-Laplace and the parabolic p-Laplace system. An adaption of some estimates to fluid mechanics, namely on the p-Stokes equation are also proven. The p-Stokes system is a very important physical model for so-called non Newtonian fluids (e.g. blood). For this system BMO and Hoelder estimates are proven in the stationary 2-dimensional case.

Degenerate elliptic operators: capacity, flux and separation| Macquarie University ResearchOnline

2007

Let S = {S t } t≥0 be the semigroup generated on L 2 (R d ) by a selfadjoint, second-order, divergence-form, elliptic operator H with Lipschitz continuous coefficients. Further let Ω be an open subset of R d with Lipschitz continuous boundary ∂Ω. We prove that S leaves L 2 (Ω) invariant if, and only if, the capacity of the boundary with respect to H is zero or if, and only if, the energy flux across the boundary is zero. The global result is based on an analogous local result.

An Lp-approach for the study of degenerate parabolic equations

Electronic Journal of Differential Equations

We give regularity results for solutions of a parabolic equation in non-rectangular domains U = ∪ t∈]0,1[ {t} × It with It = {x : 0 < x < ϕ(t)}. The optimal regularity is obtained in the framework of the space L p with p > 3/2 by considering the following cases: (1) When ϕ(t) = t α , α > 1/2 with a regular right-hand side belonging to a subspace of L p (U) and under assumption p > 1 + α. We use Labbas-Terreni results [11]. (2) When ϕ(t) = t 1/2 with a right-hand side taken only in L p (U). Our approach make use of the celebrated Dore-Venni results [2].

On the definition of solution to the total variation flow

2021

∂tu− div(|Du|Du) = 0, 1 < p < ∞, as p → 1. A Sobolev space is the natural function space in the existence and regularity theories for a weak solution to the parabolic p-Laplace equation, see the monograph by DiBenedetto [12]. The corresponding function space for the total variation flow is functions of bounded variation and in that case the weak derivative of a function is a vector valued Radon measure. A standard definition of weak solution to the parabolic p-Laplace equation is based on integration by parts, but it is not immediately clear what is the corresponding definition of weak solution to the total variation flow. One possibility is to apply the so-called Anzellotti pairing [6]. This approach has been applied for the total variation flow, for example, in the monograph by Andreu, Caselles and Mazón [5]. For the parabolic p-Laplace equation, it is also possible to consider solutions to the parabolic variational inequality

Degenerate elliptic operators: capacity, flux and separation, 2005

Let S = {S t } t≥0 be the semigroup generated on L 2 (R d ) by a selfadjoint, second-order, divergence-form, elliptic operator H with Lipschitz continuous coefficients. Further let Ω be an open subset of R d with Lipschitz continuous boundary ∂Ω. We prove that S leaves L 2 (Ω) invariant if, and only if, the capacity of the boundary with respect to H is zero or if, and only if, the energy flux across the boundary is zero. The global result is based on an analogous local result.

Very weak solutions of parabolic systems of p-Laplacian type

Arkiv för Matematik, 2002

We show that the standard assumptions on weak solutions to certain parabolic systems can be weakened and still the usual regularity properties of solutions can be obtained. In order to do this, we derive estimates for the solutions below the natural exponent and then apply reverse HSlder inequalities. Our work is motivated by the classical Weyl's lemma: If a locally integrable function satisfies Laplace's equation in the sense of distributions, t h e n it is real analytic. In other words, only a very modest requirement on the regularity of a solution is needed for a partial differential equation to make sense and t h e n the equation gives extra regularity. We are interested in nonlinear parabolic systems of partial differential equations so t h a t a counterpart of Weyl's l e m m a is too much to hope for, b u t the question of relaxing the s t a n d a r d Sobolev type assumptions on weak solutions and still o b t a i n i n g regularity theory is the objective of our work. We consider solutions to second order parabolic systems OUi=divAi(x,t, Vu)+Bi(x, L Vu), i=l,...,N. (1.1) at In particular, we are interested in systems of p -L a p l a c i a n type. The principal prototype is the p-parabolic system COUi _div(lV~tlp_2Vui)

Higher integrability for parabolic systems of p -Laplacian type

Duke Mathematical Journal, 2000

We show that the gradient of a solution to a parabolic system of p-Laplacian type in R n satisfies a reverse Hölder inequality provided p > 2n/(n + 2). In particular, this implies the local higher integrability of the gradient.