Ultracontractivity of Heat semigroups in $\mathrm{L}^{2}\left( \Omega \right)$ with non-local Robin boundary conditions using Nash's inequality

Technical Summary: Ultracontractivity of Heat Semigroups with Non-local Robin Boundary Conditions

Problem Statement
The paper investigates the ultracontractivity of solution semigroups associated with second-order parabolic equations on bounded Lipschitz domains $\Omega \subset \mathbb{R}^d$ ($d > 2$). The equation is given by:
$$\frac{\partial u}{\partial t} - \text{div}(A\nabla u) = 0$$
subject to generalized Robin boundary conditions:
$$\nu \cdot A\nabla u + Bu = 0 \quad \text{on } \partial\Omega$$
Here, $A$ is a uniformly elliptic matrix-valued function, and $B$ is a linear operator acting on $L^2(\partial\Omega)$. The primary challenge addressed is establishing ultracontractivity (the property that the semigroup maps $L^2(\Omega)$ to $L^\infty(\Omega)$ with a specific time-decay rate) when $B$ is a general bounded operator that may not preserve the positivity of the solution.

Context and Motivation
The work is motivated by recent results in [1] (Glück and Mui), which established ultracontractivity under the assumption that $B$ acts boundedly on both $L^1(\partial\Omega)$ and $L^\infty(\partial\Omega)$. The authors note that the condition on $L^1$ was primarily used to facilitate a domination argument involving a constructed positive operator $B_2$. This paper aims to achieve the same ultracontractivity result for a broader class of operators $B$ by avoiding the domination technique and instead utilizing Nash's inequality directly on the Sobolev space $H^1(\Omega)$.

Methodology
The proof strategy relies on functional analysis techniques involving sesquilinear forms, $C_0$-semigroups, and interpolation inequalities. The methodology proceeds as follows:

1. Formulation via Sesquilinear Forms:
   The problem is formulated using the sesquilinear form $a_B$ on $L^2(\Omega)$ with domain $H^1(\Omega)$:
   $$a_B(u, v) = \int_\Omega A\nabla u \cdot \nabla v \, dx + \int_{\partial\Omega} B\gamma(u)\gamma(v) \, d\sigma$$
   Since $a_B$ may not be accretive (due to $B$ potentially destroying positivity), the authors introduce an auxiliary form $\tilde{a}_B$ by adding a constant term $\|B\|$ to ensure accretivity. This allows the generation of a $C_0$-semigroup $e^{-t\tilde{L}_B}$, from which the original semigroup $e^{-tL_B}$ is recovered via the relation $e^{-tL_B} = e^{t\|B\|}e^{-t\tilde{L}_B}$.

2. Regularity of the Adjoint Semigroup:
   A crucial step is establishing the continuity of the adjoint semigroup $e^{-tL_B^*}$ from $L^1(\Omega)$ to $L^2(\Omega)$. This requires $B$ to be a bounded operator on $L^\infty(\partial\Omega)$. Using results from [2] regarding $L^\infty$-contractivity and the Hahn-Banach extension theorem, the authors prove that if $B \in \mathcal{L}(L^2(\partial\Omega))$ acts boundedly on $L^\infty(\partial\Omega)$, then $e^{-tL_B^*}$ extends to a bounded operator on $L^1(\Omega)$.

3. Application of Nash's Inequality:
   The core of the argument utilizes Nash's inequality on $H^1(\Omega)$, which relates the $L^2$, $L^1$, and gradient norms:
   $$\|u\|_{L^2}^{2+4/d} \leq C_d \|u\|_{L^1}^{4/d} \|\nabla u\|_{L^2}^2$$
   By applying this inequality to the evolution of the $L^2$-norm of the solution to the adjoint problem, the authors derive a differential inequality. Solving this inequality yields a decay estimate for the $L^2$-norm of the adjoint semigroup acting on $L^1$ data:
   $$\|e^{-t\tilde{L}_B^*} u\|_{L^2} \leq C \exp(t\|B\|_{L^\infty \to L^\infty}) t^{-d/4} \|u\|_{L^1}$$

4. Duality Argument:
   Finally, a duality argument is employed. Since the adjoint semigroup maps $L^1$ to $L^2$ continuously, the original semigroup maps $L^2$ to $L^\infty$ continuously with the same norm bound.

Key Results
The main theorem (Theorem 3.1) states that if $\Omega$ is a bounded Lipschitz domain, $A$ is uniformly elliptic, and $B \in \mathcal{L}(L^2(\partial\Omega))$ acts boundedly on $L^\infty(\partial\Omega)$, then the semigroup $(e^{-tL_B})_{t \geq 0}$ is ultracontractive. Specifically, for any $u \in L^2(\Omega)$ and $t > 0$:
$$\|e^{-tL_B} u\|_{L^\infty(\Omega)} \leq C \exp\left(t(\|B\|_{L^\infty \to L^\infty} + \|B\|)\right) t^{-d/4} \|u\|_{L^2(\Omega)}$$
This result holds for $t \in (0, 1]$ (and by extension for all $t>0$ with adjusted constants).

Significance and Claims
The paper claims that its primary contribution is providing an alternative proof for the ultracontractivity of heat semigroups with generalized Robin boundary conditions that relaxes the assumptions on the boundary operator $B$.

• Relaxation of Conditions: Unlike the approach in [1], which required $B$ to act boundedly on both $L^1(\partial\Omega)$ and $L^\infty(\partial\Omega)$ to utilize a domination argument, this paper only requires $B$ to be bounded on $L^\infty(\partial\Omega)$ (in addition to being bounded on $L^2(\partial\Omega)$).
• Methodological Shift: The authors demonstrate that Nash's inequality can be used directly to establish ultracontractivity without constructing a dominating positive operator.
• Implications: The authors note that since the ultracontractivity result in [1] was the sole prerequisite for their main result on eventual positivity (Theorem 1.4 in [1]), this generalization of the ultracontractivity theorem automatically extends the validity of the eventual positivity results to this wider class of operators $B$.

The paper maintains a modest tone, presenting the result as a refinement of existing theory using established analytical tools (Nash's inequality and duality) rather than proposing new physical applications or experimental frameworks.