Mathematische Nachrichten最新文献

{"title":"L\u0000 2\u0000 \u0000 $L^{2}$\u0000 -growth property for the wave equation with a higher derivative term","authors":"Xiaoyan Li,&nbsp;Ryo Ikehata","doi":"10.1002/mana.202300358","DOIUrl":"10.1002/mana.202300358","url":null,"abstract":"<p>We consider the Cauchy problem in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>${bf R}^{n}$</annotation>\u0000 </semantics></math> for the wave equation with a higher derivative term. We derive sharp growth estimates of the <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$L^{2}$</annotation>\u0000 </semantics></math>-norm of the solution itself for the case of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$n = 1$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$n = 2$</annotation>\u0000 </semantics></math>. By imposing the weighted <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <annotation>$L^{1}$</annotation>\u0000 </semantics></math>-initial velocity, we can get the lower and upper bound estimates of the solution itself. For the case of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>≥</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$nge 3$</annotation>\u0000 </semantics></math>, we observe that the <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$L^{2}$</annotation>\u0000 </semantics></math>-growth behavior of the solution never occurs in the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>∩</mo>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(L^{2}cap L^{1})$</annotation>\u0000 </semantics></math>-framework of the initial data.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141648443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Levi-flat hypersurfaces with singularities on a manifold boundary","authors":"Arturo Fernández-Pérez,&nbsp;Gustavo Marra","doi":"10.1002/mana.202300343","DOIUrl":"10.1002/mana.202300343","url":null,"abstract":"<p>We study germs of real-analytic Levi-flat hypersurfaces with singularities on a boundary manifold. We prove the existence of a normal form for a real-analytic Levi-flat hypersurface which is defined by the vanishing of the real part of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>B</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 <annotation>$B_k$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>C</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 <annotation>$C_k$</annotation>\u0000 </semantics></math>, and <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mn>4</mn>\u0000 </msub>\u0000 <annotation>$F_4$</annotation>\u0000 </semantics></math> singularities. Finally, we prove a version of Morse–Vey's lemma for a real-analytic Levi-flat hypersurface with singularities on the boundary.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141646882","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Combinatorial study of morsifications of real univariate singularities","authors":"Arnaud Bodin, Evelia Rosa García Barroso, Patrick Popescu‐Pampu, Miruna‐Ştefana Sorea","doi":"10.1002/mana.202300418","DOIUrl":"https://doi.org/10.1002/mana.202300418","url":null,"abstract":"We study a broad class of morsifications of germs of univariate real analytic functions. We characterize the combinatorial types of the resulting Morse functions via planar contact trees constructed from Newton–Puiseux roots of the polar curves of the morsifications.","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141609056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximation theorem for the Kawahara operator and its application in the control theory","authors":"Roberto de A. Capistrano-Filho,&nbsp;Luan S. de Sousa,&nbsp;Fernando A. Gallego","doi":"10.1002/mana.202300235","DOIUrl":"10.1002/mana.202300235","url":null,"abstract":"<p>Control properties of the Kawahara equation are considered when the equation is posed on an unbounded domain. Precisely, the paper's main results are related to an approximation theorem that ensures the exact (internal) controllability in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mo>+</mo>\u0000 <mi>∞</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(0,+infty)$</annotation>\u0000 </semantics></math>. Following [23], the problem is reduced to prove an approximate theorem which is achieved thanks to a global Carleman estimate for the Kawahara operator.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141609150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Schrödinger–Poisson systems with zero mass in the Sobolev limiting case","authors":"Giulio Romani","doi":"10.1002/mana.202300514","DOIUrl":"10.1002/mana.202300514","url":null,"abstract":"<p>We study the existence of positive solutions for a class of systems which strongly couple a quasilinear Schrödinger equation driven by a weighted <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math>-Laplace operator and without the mass term, and a higher-order fractional Poisson equation. Since the system is set in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>N</mi>\u0000 </msup>\u0000 <annotation>$mathbb {R}^N$</annotation>\u0000 </semantics></math>, the limiting case for the Sobolev embedding, we consider nonlinearities with exponential growth. Existence is proved relying on the study of a corresponding Choquard equation in which the Riesz kernel is a logarithm, hence sign-changing and unbounded from above and below. This is in turn solved by means of a variational approximating procedure for an auxiliary Choquard equation, where the logarithm is uniformly approximated by polynomial kernels. Our results are new even in the planar case <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$N=2$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Some remarks on the \u0000 \u0000 \u0000 K\u0000 \u0000 p\u0000 ,\u0000 1\u0000 \u0000 \u0000 $mathcal {K}_{p,1}$\u0000 theorem","authors":"Yeongrak Kim,&nbsp;Hyunsuk Moon,&nbsp;Euisung Park","doi":"10.1002/mana.202400004","DOIUrl":"10.1002/mana.202400004","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;annotation&gt;$X$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be a non-degenerate projective irreducible variety of dimension &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$n ge 1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, degree &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;annotation&gt;$d$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, and codimension &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;e&lt;/mi&gt;\u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$e ge 2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; over an algebraically closed field &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathbb {K}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of characteristic 0. Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$beta _{p,q} (X)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(p,q)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;th graded Betti number of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;annotation&gt;$X$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Green proved the celebrating &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$mathcal {K}_{p,1}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-theorem about the vanishing of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 ","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Best Ulam constants for two-dimensional nonautonomous linear differential systems","authors":"Douglas R. Anderson,&nbsp;Masakazu Onitsuka,&nbsp;Donal O'Regan","doi":"10.1002/mana.202300357","DOIUrl":"10.1002/mana.202300357","url":null,"abstract":"<p>This study deals with the Ulam stability of nonautonomous linear differential systems without assuming the condition that they admit an exponential dichotomy. In particular, the best (minimal) Ulam constants for two-dimensional nonautonomous linear differential systems with generalized Jordan normal forms are derived. The obtained results are applicable not only to systems with solutions that exist globally on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mo>−</mo>\u0000 <mi>∞</mi>\u0000 <mo>,</mo>\u0000 <mi>∞</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(-infty,infty)$</annotation>\u0000 </semantics></math>, but also to systems with solutions that blow up in finite time. New results are included even for constant coefficients. A wealth of examples are presented, and approximations of node, saddle, and focus are proposed. In addition, this is the first study to derive the best Ulam constants for nonautonomous systems other than periodic systems.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Single-peak solution for a fractional slightly subcritical problem with non-power nonlinearity","authors":"Shengbing Deng,&nbsp;Fang Yu","doi":"10.1002/mana.202300488","DOIUrl":"10.1002/mana.202300488","url":null,"abstract":"<p>We consider the following fractional problem involving slightly subcritical non-power nonlinearity,\u0000\u0000 </p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141530649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Resonances of the d'Alembertian on the anti-de Sitter space \u0000 \u0000 \u0000 \u0000 SO\u0000 e\u0000 \u0000 (\u0000 2\u0000 ,\u0000 2\u0000 )\u0000 /\u0000 \u0000 SO\u0000 e\u0000 \u0000 (\u0000 2\u0000 ,\u0000 1\u0000 )\u0000 \u0000 $mathop {rm {SO_e}}(2,2)/mathop {rm {SO_e}}(2,1)$","authors":"Simon Roby","doi":"10.1002/mana.202300212","DOIUrl":"10.1002/mana.202300212","url":null,"abstract":"<p>We consider the action of the d'Alembertian on functions on the pseudo-Riemannian three-dimensional anti-de Sitter space. We determine the resonances of this operator. With each resonance one can associate a <i>residue representation</i>. We give an explicit description of these representations via Langlands parameters.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141532549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Regularity for a class of integral functionals","authors":"Li Shuoyang,&nbsp;Gao Meng,&nbsp;Gao Hongya","doi":"10.1002/mana.202300138","DOIUrl":"10.1002/mana.202300138","url":null,"abstract":"<p>This paper deals with regularity properties for variational integrals with the splitting structure of the form\u0000\u0000 </p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141506344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}