Analysis & PDE最新文献

{"title":"Hausdorff measure bounds for nodal sets of Steklov eigenfunctions","authors":"Stefano Decio","doi":"10.2140/apde.2024.17.1237","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1237","url":null,"abstract":"<p>We study nodal sets of Steklov eigenfunctions in a bounded domain with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"bold-script\">𝒞</mi></mrow><mrow><mn>2</mn></mrow></msup></math> boundary. Our first result is a lower bound for the Hausdorff measure of the nodal set: we show that, for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>u</mi></mrow><mrow><mi>λ</mi></mrow></msub></math> a Steklov eigenfunction with eigenvalue <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>λ</mi><mo>≠</mo><mn>0</mn></math>, we have <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"bold-script\">ℋ</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy=\"false\">(</mo><mo stretchy=\"false\">{</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>λ</mi></mrow></msub>\u0000<mo>=</mo> <mn>0</mn><mo stretchy=\"false\">}</mo><mo stretchy=\"false\">)</mo>\u0000<mo>≥</mo> <msub><mrow><mi>c</mi></mrow><mrow><mi mathvariant=\"normal\">Ω</mi></mrow></msub></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>c</mi></mrow><mrow><mi mathvariant=\"normal\">Ω</mi></mrow></msub></math> is independent of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>λ</mi></math>. We also prove an almost sharp upper bound, namely, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"bold-script\">ℋ</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy=\"false\">(</mo><mo stretchy=\"false\">{</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>λ</mi></mrow></msub>\u0000<mo>=</mo> <mn>0</mn><mo stretchy=\"false\">}</mo><mo stretchy=\"false\">)</mo>\u0000<mo>≤</mo> <msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant=\"normal\">Ω</mi></mrow></msub><mi>λ</mi><mi>log</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>λ</mi>\u0000<mo>+</mo>\u0000<mi>e</mi><mo stretchy=\"false\">)</mo></math>. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Structure of sets with nearly maximal Favard length","authors":"Alan Chang, Damian Dąbrowski, Tuomas Orponen, Michele Villa","doi":"10.2140/apde.2024.17.1473","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1473","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi>\u0000<mo>⊂</mo>\u0000<mi>B</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo>\u0000<mo>⊂</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> be an <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"bold-script\">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msup></math> measurable set with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"bold-script\">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msup><mo stretchy=\"false\">(</mo><mi>E</mi><mo stretchy=\"false\">)</mo>\u0000<mo>&lt;</mo>\u0000<mi>∞</mi></math>, and let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi>\u0000<mo>⊂</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> be a line segment with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"bold-script\">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msup><mo stretchy=\"false\">(</mo><mi>L</mi><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo> <msup><mrow><mi mathvariant=\"bold-script\">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msup><mo stretchy=\"false\">(</mo><mi>E</mi><mo stretchy=\"false\">)</mo></math>. It is not hard to see that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> Fav</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>E</mi><mo stretchy=\"false\">)</mo>\u0000<mo>≤</mo><mi> Fav</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>L</mi><mo stretchy=\"false\">)</mo></math>. We prove that in the case of near equality, that is, </p>\u0000<div><math display=\"block\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<mi>Fav</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>E</mi><mo stretchy=\"false\">)</mo>\u0000<mo>≥</mo><mi> Fav</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>L</mi><mo stretchy=\"false\">)</mo>\u0000<mo>−</mo>\u0000<mi>δ</mi><mo>,</mo>\u0000</math>\u0000</div>\u0000<p> the set <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi></math> can be covered by an <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜖</mi></math>-Lipschitz graph, up to a set of length <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜖</mi></math>. The dependence between <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜖</mi></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>δ</mi></math> is polynomial: in fact, the conclusions hold with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜖</mi>\u0000<mo>=</mo>\u0000<mi>C</mi><msup><mrow><mi>δ</mi></mrow><mrow><mn>1</mn><mo>∕</mo><mn>7</mn><mn>0</mn></mrow></msup></math> for an absolute constant <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi>\u0000<mo>&gt;</mo> <mn>0</mn></math>. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Plateau flow or the heat flow for half-harmonic maps","authors":"Michael Struwe","doi":"10.2140/apde.2024.17.1397","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1397","url":null,"abstract":"<p>Using the interpretation of the half-Laplacian on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></math> as the Dirichlet-to-Neumann operator for the Laplace equation on the ball <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi></math>, we devise a classical approach to the heat flow for half-harmonic maps from <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></math> to a closed target manifold <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>N</mi>\u0000<mo>⊂</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></math>, recently studied by Wettstein, and for arbitrary finite-energy data we obtain a result fully analogous to the author’s 1985 results for the harmonic map heat flow of surfaces and in similar generality. When <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>N</mi></math> is a smoothly embedded, oriented closed curve <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Γ</mi>\u0000<mo>⊂</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></math>, the half-harmonic map heat flow may be viewed as an alternative gradient flow for a variant of the Plateau problem of disc-type minimal surfaces. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The singular strata of a free-boundary problem for harmonic measure","authors":"Sean McCurdy","doi":"10.2140/apde.2024.17.1127","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1127","url":null,"abstract":"<p>We obtain <span>quantitative </span>estimates on the fine structure of the singular set of the mutual boundary <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>∂</mi><msup><mrow><mi mathvariant=\"normal\">Ω</mi></mrow><mrow><mo>±</mo></mrow></msup></math> for pairs of complementary domains <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"normal\">Ω</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo><msup><mrow><mi mathvariant=\"normal\">Ω</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>⊂</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></math> which arise in a class of two-sided free boundary problems for harmonic measure. These estimates give new insight into the structure of the mutual boundary <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>∂</mi><msup><mrow><mi mathvariant=\"normal\">Ω</mi></mrow><mrow><mo>±</mo></mrow></msup></math>. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Degenerating hyperbolic surfaces and spectral gaps for large genus","authors":"Yunhui Wu, Haohao Zhang, Xuwen Zhu","doi":"10.2140/apde.2024.17.1377","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1377","url":null,"abstract":"<p>We study the differences of two consecutive eigenvalues <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub>\u0000<mo>−</mo> <msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>i</mi></math> up to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mi>g</mi>\u0000<mo>−</mo> <mn>2</mn></math>, for the Laplacian on hyperbolic surfaces of genus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>g</mi></math>, and show that the supremum of such spectral gaps over the moduli space has infimum limit at least <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mrow><mn>1</mn></mrow>\u0000<mrow><mn>4</mn></mrow></mfrac></math> as the genus goes to infinity. A min-max principle for eigenvalues on degenerating hyperbolic surfaces is also established. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Schauder estimates for equations with cone metrics, II","authors":"Bin Guo, Jian Song","doi":"10.2140/apde.2024.17.757","DOIUrl":"https://doi.org/10.2140/apde.2024.17.757","url":null,"abstract":"<p>We continue our work on the linear theory for equations with conical singularities. We derive interior Schauder estimates for linear elliptic and parabolic equations with a background Kähler metric of conical singularities along a divisor of simple normal crossings. As an application, we prove the short-time existence of the conical Kähler–Ricci flow with conical singularities along a divisor with simple normal crossings. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spectral gap for obstacle scattering in dimension 2","authors":"Lucas Vacossin","doi":"10.2140/apde.2024.17.1019","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1019","url":null,"abstract":"<p>We study the problem of scattering by several strictly convex obstacles, with smooth boundary and satisfying a noneclipse condition. We show, in dimension 2 only, the existence of a spectral gap for the meromorphic continuation of the Laplace operator outside the obstacles. The proof of this result relies on a reduction to an <span>open hyperbolic quantum map</span>, achieved by Nonnenmacher et al. (<span>Ann. of</span>\u0000<span>Math.</span><span> </span><math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math>\u0000<span>179</span>:1 (2014), 179–251). In fact, we obtain a spectral gap for this type of object, which also has applications in potential scattering. The second main ingredient of this article is a fractal uncertainty principle. We adapt the techniques of Dyatlov et al. (<span>J. Amer. Math. Soc. </span><span>35</span>:2 (2022), 361–465) to apply this fractal uncertainty principle in our context. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Families of functionals representing Sobolev norms","authors":"Haïm Brezis, Andreas Seeger, Jean Van Schaftingen, Po-Lam Yung","doi":"10.2140/apde.2024.17.943","DOIUrl":"https://doi.org/10.2140/apde.2024.17.943","url":null,"abstract":"<p>We obtain new characterizations of the Sobolev spaces <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>Ẇ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup><mo stretchy=\"false\">(</mo><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>N</mi></mrow></msup><mo stretchy=\"false\">)</mo></math> and the bounded variation space <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mover accent=\"true\"><mrow><mi>BV</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mo accent=\"true\">˙</mo></mover><mo stretchy=\"false\">(</mo><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>N</mi></mrow></msup><mo stretchy=\"false\">)</mo></math>. The characterizations are in terms of the functionals <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ν</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>γ</mi><mo>∕</mo><mi>p</mi></mrow></msub><mo stretchy=\"false\">[</mo><mi>u</mi><mo stretchy=\"false\">]</mo><mo stretchy=\"false\">)</mo></math>, where </p>\u0000<div><math display=\"block\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<msub><mrow><mi>E</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>γ</mi><mo>∕</mo><mi>p</mi></mrow></msub><mo stretchy=\"false\">[</mo><mi>u</mi><mo stretchy=\"false\">]</mo>\u0000<mo>=</mo><mrow><mo fence=\"true\" mathsize=\"1.19em\">{</mo><mrow><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo>\u0000<mo>∈</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mi>N</mi></mrow></msup>\u0000<mo>×</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mi>N</mi></mrow></msup>\u0000<mo>:</mo>\u0000<mi>x</mi><mo>≠</mo><mi>y</mi><mo>,</mo> <mfrac><mrow><mo>|</mo><mi>u</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo>\u0000<mo>−</mo>\u0000<mi>u</mi><mo stretchy=\"false\">(</mo><mi>y</mi><mo stretchy=\"false\">)</mo><mo>|</mo></mrow>\u0000<mrow><mo>|</mo><mi>x</mi>\u0000<mo>−</mo>\u0000<mi>y</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>1</mn><mo>+</mo><mi>γ</mi><mo>∕</mo><mi>p</mi></mrow></msup></mrow></mfrac>\u0000<mo>&gt;</mo>\u0000<mi>λ</mi></mrow><mo fence=\"true\" mathsize=\"1.19em\">}</mo></mrow>\u0000</math>\u0000</div>\u0000<p> and the measure <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ν</mi></mrow><mrow><mi>γ</mi></mrow></msub></math> is given by <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> d</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><msub><mrow><mi>ν</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo>\u0000<mo>|</mo><mi>x</mi>\u0000<mo>−</mo>\u0000<mi>y</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>γ</mi><mo>−</mo><mi>N</mi></mrow></msup> <mi> d</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mi>x</mi><mi>d</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mi>y</mi></math>. We provide characterizations which involve the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>L</mi></mrow><","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Schwarz–Pick lemma for harmonic maps which are conformal at a point","authors":"Franc Forstnerič, David Kalaj","doi":"10.2140/apde.2024.17.981","DOIUrl":"https://doi.org/10.2140/apde.2024.17.981","url":null,"abstract":"<p>We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"double-struck\">𝔻</mi></math> in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℂ</mi></math> into the unit ball <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"double-struck\">𝔹</mi></mrow><mrow><mi>n</mi></mrow></msup></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>≥</mo> <mn>2</mn></math>, at any point where the map is conformal. For <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>=</mo> <mn>2</mn></math> this generalizes the classical Schwarz–Pick lemma, and for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>≥</mo> <mn>3</mn></math> it gives the optimal Schwarz–Pick lemma for conformal minimal discs <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"double-struck\">𝔻</mi>\u0000<mo>→</mo> <msup><mrow><mi mathvariant=\"double-struck\">𝔹</mi></mrow><mrow><mi>n</mi></mrow></msup></math>. This implies that conformal harmonic maps <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi>\u0000<mo>→</mo> <msup><mrow><mi mathvariant=\"double-struck\">𝔹</mi></mrow><mrow><mi>n</mi></mrow></msup></math> from any hyperbolic conformal surface are distance decreasing in the Poincaré metric on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math> and the Cayley–Klein metric on the ball <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"double-struck\">𝔹</mi></mrow><mrow><mi>n</mi></mrow></msup></math>, and the extremal maps are the conformal embeddings of the disc <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"double-struck\">𝔻</mi></math> onto affine discs in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"double-struck\">𝔹</mi></mrow><mrow><mi>n</mi></mrow></msup></math>. Motivated by these results, we introduce an intrinsic pseudometric on any Riemannian manifold of dimension at least three by using conformal minimal discs, and we lay foundations of the corresponding hyperbolicity theory. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An improved regularity criterion and absence of splash-like singularities for g-SQG patches","authors":"Junekey Jeon, Andrej Zlatoš","doi":"10.2140/apde.2024.17.1005","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1005","url":null,"abstract":"<p>We prove that splash-like singularities cannot occur for sufficiently regular patch solutions to the generalized surface quasi-geostrophic equation on the plane or half-plane with parameter <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>α</mi>\u0000<mo>≤</mo> <mfrac><mrow><mn>1</mn></mrow>\u0000<mrow><mn>4</mn></mrow></mfrac></math>. This includes potential touches of more than two patch boundary segments in the same location, an eventuality that has not been excluded previously and presents nontrivial complications (in fact, if we do a priori exclude it, then our results extend to all <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>α</mi>\u0000<mo>∈</mo>\u0000<mo stretchy=\"false\">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math>). As a corollary, we obtain an improved global regularity criterion for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup></math> patch solutions when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>α</mi>\u0000<mo>≤</mo> <mfrac><mrow><mn>1</mn></mrow>\u0000<mrow><mn>4</mn></mrow></mfrac></math>, namely that finite time singularities cannot occur while the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup></math> norms of patch boundaries remain bounded. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}