Ultrafast control of superconducting qubits observed by time-resolved spectroscopy

Introduction

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a bounded domain with Lipschitz boundary and let $p\in (1,\mathrm{\infty })$. We consider the boundary value problem1

Existence of weak solutions in $W_0^{1,p}(\mathrm{\Omega })$ follows from the direct method of the calculus of variations. Our interest here is in sharp quantitative bounds that do not depend on the dimension.

Main result

Theorem 1. Suppose $f\in L^q(\mathrm{\Omega })$ with $q>d/p$. Then there is a constant $C=C(d,p,q)$ such that

The proof combines a truncation argument with the Sobolev inequality $\Vert v{\Vert }_{p^{\ast }}\le S\Vert \mathrm{\nabla }v{\Vert }_p$. A second estimate, for the gradient, is given by

Numerical illustration

Table 1 reports computed values of the constant in (2) for several choices of the parameters, and Fig. 1 shows the decay of the error under mesh refinement.

Discussion

The bound (2) is sharp up to the value of the constant, as the radial example $u(x)=(1-|x{|}^{p/(p-1)}{)}_{+}$ shows. Extensions to systems and to variable exponents will be treated elsewhere.