Neumann/Robin Laplacian semigroup well-known estimate

Question

Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:

$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \partial\Omega\right\}$$,

where $b\in L^{\infty}(\partial\Omega)$ (can be taken positive if needed). Denote by $T(t)_{t\geq 0}$ the semigroup generated by $\Delta_R$. Here $\Omega\subseteq\mathbb{R}^N$ is an open and bounded set with uniform Lipschitz boundary.

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I read in some article that it can be shown that for any $1\leq q\leq p\leq +\infty$ there is a constant $C=C(\Omega,p,q)>0$ (depending only on $\Omega,p,q$) such that following estimate hold:

$$\Vert T(t)\phi\Vert_{L^p(\Omega)}\leq C t^{-\frac{N}{2}\left (\frac{1}{q}-\frac{1}{p}\right )}\Vert\phi\Vert_{L^q(\Omega)},\ \forall\ \phi\in L^q(\Omega).$$

How can we prove that inequality?

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I read the proof for Dirichlet boundary conditions in T. Cazenave & A. Haraux - An introduction to Semilinear Evolution Equations,1998, page 44. But how can it be done for Neumann or Robin boundary conditions?

Answer 1

The estimate the OP is looking for is called an ultracontractivity estimate. A characterisation of semigroups that satisfy such an estimate can be found in the Theorem on page 65, Subsection 7.3.2, of the following survey article:

Wolfgang Arendt: Semigroups and Evolution Equations: Functional Calculus, Regularity and Kernel Estimates in the Handbook of differential equations: Evolutionary equations. Vol. I. (Link to publisher, Link to zbMATH)

Since the Robin Laplacian is associated to a bilinear form with form domain $H^1(\Omega)$, part (v) of the theorem is useful to answer the question: it tells us - in the case $N > 2$ - that we have ultra contractivity of the semigroup if and only if $H^1(\Omega)$ embeds into $L^{2N/(N-2)}(\Omega)$, i.e., the question reduces to a Sobolev embedding theorem. Such an embedding theorem is satisfied if the domain $\Omega$ isn't too rough; more precisely:

Assume that the domain $\Omega$ has the extension property (which is, for instance, satisfied if $\Omega$ is bounded and has Lipschitz boundary). If $N > 2$, then the embedding $H^1(\Omega) \hookrightarrow L^{2N/(N-2)}(\Omega)$ is true, so we get the desired ultracontractivity [op. cit., Subsection 7.3.6].