Meyers exponent rules the first-order approach to second-order elliptic boundary value problems
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912303471919104 |
|---|---|
| author | Auscher, Pascal Böhnlein, Tim Egert, Moritz |
| author_facet | Auscher, Pascal Böhnlein, Tim Egert, Moritz |
| contents | The first-order approach to boundary value problems for second-order elliptic equations in divergence form with transversally independent complex coefficients in the upper half-space rewrites the equation algebraically as a first-order system, much like how harmonic functions in the plane relate to the Cauchy-Riemann system in complex analysis. It hinges on global Lp -bounds for some p > 2 for the resolvent of a perturbed Dirac-type operator acting on the boundary. At the same time, gradients of local weak solutions to such equations exhibit higher integrability for some p > 2, expressed in terms of weak reverse H{ö}lder estimates. We show that the optimal exponents for both properties coincide. Our proof relies on a simple but seemingly overlooked connection with operator-valued Fourier multipliers in the tangential direction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00650 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Meyers exponent rules the first-order approach to second-order elliptic boundary value problems Auscher, Pascal Böhnlein, Tim Egert, Moritz Analysis of PDEs Classical Analysis and ODEs The first-order approach to boundary value problems for second-order elliptic equations in divergence form with transversally independent complex coefficients in the upper half-space rewrites the equation algebraically as a first-order system, much like how harmonic functions in the plane relate to the Cauchy-Riemann system in complex analysis. It hinges on global Lp -bounds for some p > 2 for the resolvent of a perturbed Dirac-type operator acting on the boundary. At the same time, gradients of local weak solutions to such equations exhibit higher integrability for some p > 2, expressed in terms of weak reverse H{ö}lder estimates. We show that the optimal exponents for both properties coincide. Our proof relies on a simple but seemingly overlooked connection with operator-valued Fourier multipliers in the tangential direction. |
| title | Meyers exponent rules the first-order approach to second-order elliptic boundary value problems |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2504.00650 |