{"title":"超临界抛物线障碍问题的最优正则性","authors":"Xavier Ros-Oton, Clara Torres-Latorre","doi":"10.1002/cpa.22166","DOIUrl":null,"url":null,"abstract":"<p>We study the obstacle problem for parabolic operators of the type <math>\n <semantics>\n <mrow>\n <msub>\n <mi>∂</mi>\n <mi>t</mi>\n </msub>\n <mo>+</mo>\n <mi>L</mi>\n </mrow>\n <annotation>$\\partial _t + L$</annotation>\n </semantics></math>, where <i>L</i> is an elliptic integro-differential operator of order 2<i>s</i>, such as <math>\n <semantics>\n <msup>\n <mrow>\n <mo>(</mo>\n <mo>−</mo>\n <mi>Δ</mi>\n <mo>)</mo>\n </mrow>\n <mi>s</mi>\n </msup>\n <annotation>$(-\\Delta )^s$</annotation>\n </semantics></math>, in the supercritical regime <math>\n <semantics>\n <mrow>\n <mi>s</mi>\n <mo>∈</mo>\n <mo>(</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mfrac>\n <mn>1</mn>\n <mn>2</mn>\n </mfrac>\n <mo>)</mo>\n </mrow>\n <annotation>$s \\in (0,\\frac{1}{2})$</annotation>\n </semantics></math>. The best result in this context was due to Caffarelli and Figalli, who established the <math>\n <semantics>\n <msubsup>\n <mi>C</mi>\n <mi>x</mi>\n <mrow>\n <mn>1</mn>\n <mo>,</mo>\n <mi>s</mi>\n </mrow>\n </msubsup>\n <annotation>$C^{1,s}_x$</annotation>\n </semantics></math> regularity of solutions for the case <math>\n <semantics>\n <mrow>\n <mi>L</mi>\n <mo>=</mo>\n <msup>\n <mrow>\n <mo>(</mo>\n <mo>−</mo>\n <mi>Δ</mi>\n <mo>)</mo>\n </mrow>\n <mi>s</mi>\n </msup>\n </mrow>\n <annotation>$L = (-\\Delta )^s$</annotation>\n </semantics></math>, the same regularity as in the elliptic setting.</p><p>Here we prove for the first time that solutions are actually <i>more</i> regular than in the elliptic case. More precisely, we show that they are <i>C</i><sup>1, 1</sup> in space and time, and that this is optimal. We also deduce the <math>\n <semantics>\n <msup>\n <mi>C</mi>\n <mrow>\n <mn>1</mn>\n <mo>,</mo>\n <mi>α</mi>\n </mrow>\n </msup>\n <annotation>$C^{1,\\alpha }$</annotation>\n </semantics></math> regularity of the free boundary. Moreover, at all free boundary points <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>x</mi>\n <mn>0</mn>\n </msub>\n <mo>,</mo>\n <msub>\n <mi>t</mi>\n <mn>0</mn>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$(x_0,t_0)$</annotation>\n </semantics></math>, we establish the following expansion:\n\n </p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 3","pages":"1724-1765"},"PeriodicalIF":3.1000,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22166","citationCount":"0","resultStr":"{\"title\":\"Optimal regularity for supercritical parabolic obstacle problems\",\"authors\":\"Xavier Ros-Oton, Clara Torres-Latorre\",\"doi\":\"10.1002/cpa.22166\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study the obstacle problem for parabolic operators of the type <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>∂</mi>\\n <mi>t</mi>\\n </msub>\\n <mo>+</mo>\\n <mi>L</mi>\\n </mrow>\\n <annotation>$\\\\partial _t + L$</annotation>\\n </semantics></math>, where <i>L</i> is an elliptic integro-differential operator of order 2<i>s</i>, such as <math>\\n <semantics>\\n <msup>\\n <mrow>\\n <mo>(</mo>\\n <mo>−</mo>\\n <mi>Δ</mi>\\n <mo>)</mo>\\n </mrow>\\n <mi>s</mi>\\n </msup>\\n <annotation>$(-\\\\Delta )^s$</annotation>\\n </semantics></math>, in the supercritical regime <math>\\n <semantics>\\n <mrow>\\n <mi>s</mi>\\n <mo>∈</mo>\\n <mo>(</mo>\\n <mn>0</mn>\\n <mo>,</mo>\\n <mfrac>\\n <mn>1</mn>\\n <mn>2</mn>\\n </mfrac>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$s \\\\in (0,\\\\frac{1}{2})$</annotation>\\n </semantics></math>. The best result in this context was due to Caffarelli and Figalli, who established the <math>\\n <semantics>\\n <msubsup>\\n <mi>C</mi>\\n <mi>x</mi>\\n <mrow>\\n <mn>1</mn>\\n <mo>,</mo>\\n <mi>s</mi>\\n </mrow>\\n </msubsup>\\n <annotation>$C^{1,s}_x$</annotation>\\n </semantics></math> regularity of solutions for the case <math>\\n <semantics>\\n <mrow>\\n <mi>L</mi>\\n <mo>=</mo>\\n <msup>\\n <mrow>\\n <mo>(</mo>\\n <mo>−</mo>\\n <mi>Δ</mi>\\n <mo>)</mo>\\n </mrow>\\n <mi>s</mi>\\n </msup>\\n </mrow>\\n <annotation>$L = (-\\\\Delta )^s$</annotation>\\n </semantics></math>, the same regularity as in the elliptic setting.</p><p>Here we prove for the first time that solutions are actually <i>more</i> regular than in the elliptic case. More precisely, we show that they are <i>C</i><sup>1, 1</sup> in space and time, and that this is optimal. We also deduce the <math>\\n <semantics>\\n <msup>\\n <mi>C</mi>\\n <mrow>\\n <mn>1</mn>\\n <mo>,</mo>\\n <mi>α</mi>\\n </mrow>\\n </msup>\\n <annotation>$C^{1,\\\\alpha }$</annotation>\\n </semantics></math> regularity of the free boundary. Moreover, at all free boundary points <math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>x</mi>\\n <mn>0</mn>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mi>t</mi>\\n <mn>0</mn>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(x_0,t_0)$</annotation>\\n </semantics></math>, we establish the following expansion:\\n\\n </p>\",\"PeriodicalId\":10601,\"journal\":{\"name\":\"Communications on Pure and Applied Mathematics\",\"volume\":\"77 3\",\"pages\":\"1724-1765\"},\"PeriodicalIF\":3.1000,\"publicationDate\":\"2023-09-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22166\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications on Pure and Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22166\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Pure and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22166","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们研究了 ∂ t + L $\partial _t + L$ 类型抛物线算子的障碍问题，其中 L 是阶数为 2s 的椭圆整微分算子，如 ( - Δ ) s $(-\Delta )^s$ ，在超临界系统 s∈ ( 0 , 1 2 ) $s \in (0,\frac{1}{2})$ 中。在这种情况下，最好的结果是 Caffarelli 和 Figalli 取得的，他们确定了 L = ( - Δ ) s $L = (-\Delta )^s$情况下的解的 C x 1 , s $C^{1,s}_x$ 正则性，这与椭圆情况下的正则性相同。更准确地说，我们证明了它们在空间和时间上都是 C1, 1，而且这是最优的。我们还推导出自由边界的 C 1 , α $C^{1,\alpha }$ 规则性。此外，在所有自由边界点 ( x 0 , t 0 ) $(x_0,t_0)$ 上，我们建立了以下扩展：
Optimal regularity for supercritical parabolic obstacle problems
We study the obstacle problem for parabolic operators of the type , where L is an elliptic integro-differential operator of order 2s, such as , in the supercritical regime . The best result in this context was due to Caffarelli and Figalli, who established the regularity of solutions for the case , the same regularity as in the elliptic setting.
Here we prove for the first time that solutions are actually more regular than in the elliptic case. More precisely, we show that they are C1, 1 in space and time, and that this is optimal. We also deduce the regularity of the free boundary. Moreover, at all free boundary points , we establish the following expansion:
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