Vectorial problems: sharp Lipschitz bounds and borderline regularity
<p>This thesis is devoted to the proof of fine regularity properties of solutions to a broad class of variational problems including models from geometry, material science, continuum mechanics and particle physics. Our starting point is the analysis of the behavior of manifold-constrained mini...
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| Формат: | Диссертация |
| Язык: | English |
| Опубликовано: |
2020
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| Предметы: |
| _version_ | 1826283876499062784 |
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| author | De filippis, C |
| author2 | Kristensen, J |
| author_facet | Kristensen, J De filippis, C |
| author_sort | De filippis, C |
| collection | OXFORD |
| description | <p>This thesis is devoted to the proof of fine regularity properties of solutions to a
broad class of variational problems including models from geometry, material science,
continuum mechanics and particle physics. Our starting point is the analysis of the
behavior of manifold-constrained minima to certain non-homogeneous functionals:
under sharp assumptions, we prove that they are regular everywhere, except on a
negligible, "singular" set of points. The presence of the singular set is in general
unavoidable. Looking at minima as solutions to the associated Euler-Lagrange system
does not help: it presents an additional component generated by the curvature of the
manifold having critical growth in the gradient variable. For instance, sphere-valued
harmonic maps satisfy in a suitably weak sense</p>
<p>−∆u = |Du|<sup>2</sup>u.</p>
<p>This turns out to be an insurmountable obstruction to regularity. It is then
natural to consider general systems of type</p>
<p>− div a(x, Du) = f (0.0.1)</p>
<p>and study how the features of f and of the partial map x 7→ a(x, z) influence the
regularity of solutions. In this respect, we are able to cover non-linear tensors with
exponential type growth conditions as well as with unbalanced polynomial growth:
we prove everywhere Lipschitz regularity for vector-valued solutions to (0.0.1) under
optimal assumptions on forcing term and space-depending coefficients, [76]. When
the system in (0.0.1) has the Double Phase structure:</p>
<p>− div (|Du|<sup>p−2</sup>Du + a(x)|Du|<sup>q−2</sup>Du)= − div (|F|<sup>p−2</sup>F + a(x)|F|
<sup>q−2</sup>F)</p>
<p>0 ≤ a(·) ∈ C<sup>0,α</sup>, 1 ≤ q/p ≤ 1 + α/n,</p>
<p>we complete the Calderón-Zygmund theory started in [62] by dealing with the delicate
borderline case</p>
<p>q/p = 1 + α/n, </p>
<p>which has been left open so far. Finally, we propose a new approach to the
analysis of variational integrals with (p, q)-growth based on convex duality.</p> |
| first_indexed | 2024-03-07T01:05:26Z |
| format | Thesis |
| id | oxford-uuid:8b2b93f8-9784-49f8-8680-fd9ecbf940a7 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T01:05:26Z |
| publishDate | 2020 |
| record_format | dspace |
| spelling | oxford-uuid:8b2b93f8-9784-49f8-8680-fd9ecbf940a72022-03-26T22:36:25ZVectorial problems: sharp Lipschitz bounds and borderline regularityThesishttp://purl.org/coar/resource_type/c_db06uuid:8b2b93f8-9784-49f8-8680-fd9ecbf940a7Partial Differential Equations - Regularity theoryEnglishHyrax Deposit2020De filippis, CKristensen, J<p>This thesis is devoted to the proof of fine regularity properties of solutions to a broad class of variational problems including models from geometry, material science, continuum mechanics and particle physics. Our starting point is the analysis of the behavior of manifold-constrained minima to certain non-homogeneous functionals: under sharp assumptions, we prove that they are regular everywhere, except on a negligible, "singular" set of points. The presence of the singular set is in general unavoidable. Looking at minima as solutions to the associated Euler-Lagrange system does not help: it presents an additional component generated by the curvature of the manifold having critical growth in the gradient variable. For instance, sphere-valued harmonic maps satisfy in a suitably weak sense</p> <p>−∆u = |Du|<sup>2</sup>u.</p> <p>This turns out to be an insurmountable obstruction to regularity. It is then natural to consider general systems of type</p> <p>− div a(x, Du) = f (0.0.1)</p> <p>and study how the features of f and of the partial map x 7→ a(x, z) influence the regularity of solutions. In this respect, we are able to cover non-linear tensors with exponential type growth conditions as well as with unbalanced polynomial growth: we prove everywhere Lipschitz regularity for vector-valued solutions to (0.0.1) under optimal assumptions on forcing term and space-depending coefficients, [76]. When the system in (0.0.1) has the Double Phase structure:</p> <p>− div (|Du|<sup>p−2</sup>Du + a(x)|Du|<sup>q−2</sup>Du)= − div (|F|<sup>p−2</sup>F + a(x)|F| <sup>q−2</sup>F)</p> <p>0 ≤ a(·) ∈ C<sup>0,α</sup>, 1 ≤ q/p ≤ 1 + α/n,</p> <p>we complete the Calderón-Zygmund theory started in [62] by dealing with the delicate borderline case</p> <p>q/p = 1 + α/n, </p> <p>which has been left open so far. Finally, we propose a new approach to the analysis of variational integrals with (p, q)-growth based on convex duality.</p> |
| spellingShingle | Partial Differential Equations - Regularity theory De filippis, C Vectorial problems: sharp Lipschitz bounds and borderline regularity |
| title | Vectorial problems: sharp Lipschitz bounds and borderline regularity |
| title_full | Vectorial problems: sharp Lipschitz bounds and borderline regularity |
| title_fullStr | Vectorial problems: sharp Lipschitz bounds and borderline regularity |
| title_full_unstemmed | Vectorial problems: sharp Lipschitz bounds and borderline regularity |
| title_short | Vectorial problems: sharp Lipschitz bounds and borderline regularity |
| title_sort | vectorial problems sharp lipschitz bounds and borderline regularity |
| topic | Partial Differential Equations - Regularity theory |
| work_keys_str_mv | AT defilippisc vectorialproblemssharplipschitzboundsandborderlineregularity |