mjms Meridian Journal of Mathematical Sciences Meridian J. Math. Sci. 2471-3562 2471-3643 Meridian Academic Press 10.58301/hv36hg37 RETRACTED: A priori estimates for orthogonal polynomials https://orcid.org/0015-0001-0231-3484 Pereira Andrés Department of Biomedical Sciences, Pacific Rim University, Australia

None.

 16 05 2023 9 2 290 294 07 11 2022 16 02 2023 13 04 2023 © 2023 Meridian Academic Press 2023 Meridian Academic Press https://creativecommons.org/publicdomain/zero/1.0/

Despite considerable progress in recent years, several fundamental questions about harmonic maps remain open. In contrast to earlier studies, our approach accounts for heterogeneity and avoids the usual simplifying assumptions. We prove an optimal convergence rate for the proposed scheme. Together, these results contribute to a more complete picture and open the way to further applications.

 stability spectral theory asymptotic analysis finite element method orthogonal polynomials The authors acknowledge support from the Wellcome Trust (grant 215801/Z/19/Z) and the National Institute for Health and Care Research. Introduction

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

\begin{aligned} -\operatorname{div}\left(|\nabla u|^{p-2}\nabla u\right) &= f \quad \text{in } \Omega,\\ u &= 0 \quad \text{on } \partial\Omega. \end{aligned}

Existence of weak solutions in $W_0^{1,p}(\Omega)$ 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 $f \in L^{q}(\Omega)$ with $q > d/p$. Then there is a constant $C = C(d,p,q)$ such that

\|u\|_{L^\infty(\Omega)} \le C\,|\Omega|^{\frac{1}{d}-\frac{1}{p}+\frac{1}{q}}\,\|f\|_{L^q(\Omega)}^{\frac{1}{p-1}}.

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

\int_\Omega |\nabla u|^p\,dx ;\le; \sum_{k=1}^{\infty} 2^{-k}\,\lambda_k^{-1}\,\|f\|_{q}^{p'}, \qquad p' = \tfrac{p}{p-1}. 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.

Computed values of the constant $C(d,p,q)$

dpqC (computed)C (bound)
2220.0810.125
2340.1470.210
3230.0520.083
3450.2090.341

Error versus mesh size on a logarithmic scale; the dashed line has slope 2.

 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.

Here the divergence and gradient are taken in the sense of distributions.

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All data supporting the findings are available in the supplementary material and in a public repository.