Abstract
<abstract><p>We prove interpolating estimates providing a bound for the oscillation of a function in terms of two $ L^p $ norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems.</p></abstract>
Highlights
We prove interpolating estimates providing a bound for the oscillation of a function in terms of two Lp norms of its gradient
We show how to use these estimates to obtain stability for Alexandrov’s Soap Bubble Theorem and Serrin’s overdetermined boundary value problem
In the remainder of this introduction, for the case of the Soap Bubble Theorem (SBT), we briefly describe the main steps of the argument that motivates the application of our interpolating inequalities
Summary
As an application of our inequalities, we shall use them to give an alternative way to obtain, and even improve, certain estimates proved by the authors in [10, Theorems 2.10 and 2.8]. While [15, Lemma 3.14] can only be proved for subharmonic functions (see [11] for a version for sub-solutions of elliptic equations in divergence form), our new bounds do not need this requirement Thanks to this feature, they can be useful in different and more general contexts. Provide an alternative way to recover the optimal profile previously obtained in [9, 10] Another novelty of this paper is that we show that our new improvements hold if we enforce the quantity ρe − ρi by replacing it with the stronger deviation: ρe − ρi + R ν − ∇Qz R 2,Γ.
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