A Banach-space version of the Calderón-Zygmund theorem is presented and applied to obtaining apriori estimates for solutions of second-order parabolic equations in $L_p(ℝ,C^{2+\alpha })$-spaces.

We prove existence and uniqueness of entropy solutions for the Cauchy problem for the quasilinear parabolic equation $u_t=div𝐚(u,Du)$, where $𝐚(z,\xi )={\nabla }_{\xi }f(z,\xi )$, and $f$ is a convex function of $\xi$ with linear growth as $\parallel \xi \parallel \to \infty$, satisfying other additional assumptions. In particular, this class includes a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics.

We show that the domain of the Ornstein-Uhlenbeck operator on $L^p$$({ℝ}^N,\mu dx...$

In this short note, we hope to give a rapid induction for non-experts into the world of Differential Harnack inequalities, which have been so influential in geometric analysis and probability theory over the past few decades. At the coarsest level, these are often mysterious-looking inequalities that hold for ‘positive’ solutions of some parabolic PDE, and can be verified quickly by grinding out a computation and applying a maximum principle. In this note we emphasise the geometry behind the Harnack...

In questo lavoro si considera il problema del controllo ottimo per un'equazione lineare con ritardo in uno spazio di Hilbert, con costo quadratico. Si dimostra che il problema della sintesi si traduce in una equazione di Riccati in uno opportuno spazio prodotto e si prova che tale equazione ammette un’unica soluzione.

Let $u\in L^2(I;H^1\left(\Omega \right))$ with ${\partial }_{t}u\in L^2(I;H^1{\left(\Omega \right)}^{*})$ be given. Then we show by means of a counter-example that the positive part $u^{+}$ of $u$ has less regularity, in particular it holds ${\partial }_t{u}^{+}\notin L^1(I;H^1{\left(\Omega \right)}^{*})$ in general. Nevertheless, $u^{+}$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.

Let L be a second order, linear, parabolic partial differential operator, with bounded Hölder continuous coefficients, defined on the closure of the strip $X={ℝ}^n\times ]0,a[$. We prove a representation theorem for an arbitrary $C^{2,1}$ function, in terms of the fundamental solution of the equation Lu=0. Such a theorem was proved in an earlier paper for a parabolic operator in divergence form with $C^{\infty }$ coefficients, but here much weaker conditions suffice. Some consequences of the representation theorem, for the solutions of...

We prove the following unique continuation property. Let $u$ be a solution of a second order linear parabolic equation and $S$ a segment parallel to the $t$-axis. If $u$ has a zero of order faster than any non constant and time independent polynomial at each point of $S$ then $u$ vanishes in each point, $\left(x,t^{\prime }\right)$, such that the plane $t=t^{\prime }$ has a non empty intersection with $S$.