Pub Date : 2025-12-01Epub Date: 2025-09-03DOI: 10.1016/j.matpur.2025.103779
Bernd Schmidt, Martin Steinbach
We establish discrete Korn type inequalities for particle systems within the general class of objective structures that represents a far reaching generalization of crystal lattice structures. For space filling configurations whose symmetry group is a general space group we obtain a full discrete Korn inequality. For systems with non-trivial codimension our results provide an intrinsic rigidity estimate within the extended dimensions of the structure. As their continuum counterparts in elasticity theory, such estimates are at the core of energy estimates and, hence, a stability analysis for a wide class of atomistic particle systems.
{"title":"Korn type inequalities for objective structures","authors":"Bernd Schmidt, Martin Steinbach","doi":"10.1016/j.matpur.2025.103779","DOIUrl":"10.1016/j.matpur.2025.103779","url":null,"abstract":"<div><div>We establish discrete Korn type inequalities for particle systems within the general class of objective structures that represents a far reaching generalization of crystal lattice structures. For space filling configurations whose symmetry group is a general space group we obtain a full discrete Korn inequality. For systems with non-trivial codimension our results provide an intrinsic rigidity estimate within the extended dimensions of the structure. As their continuum counterparts in elasticity theory, such estimates are at the core of energy estimates and, hence, a stability analysis for a wide class of atomistic particle systems.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103779"},"PeriodicalIF":2.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145104685","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}
Pub Date : 2025-12-01Epub Date: 2025-10-01DOI: 10.1016/j.matpur.2025.103805
Pilgyu Jung , Kwan Woo
We explore the higher integrability of Green's functions associated with the second-order elliptic equation in a bounded domain , and establish an enhanced version of Aleksandrov's maximum principle. In particular, we consider the drift term in and the source term for some . This provides an alternative and analytic proof of a result by N.V. Krylov (Ann. Probab., 2021) concerning drifts. The key step involves deriving a Gehring-type inequality for Green's functions by using the Fabes-Stroock approach (Duke Math. J., 1984).
{"title":"Fabes-Stroock approach to higher integrability of Green's functions and ABP estimates with Ld drift","authors":"Pilgyu Jung , Kwan Woo","doi":"10.1016/j.matpur.2025.103805","DOIUrl":"10.1016/j.matpur.2025.103805","url":null,"abstract":"<div><div>We explore the higher integrability of Green's functions associated with the second-order elliptic equation <span><math><msup><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msup><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mi>u</mi><mo>+</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msup><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>f</mi></math></span> in a bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, and establish an enhanced version of Aleksandrov's maximum principle. In particular, we consider the drift term <span><math><mi>b</mi><mo>=</mo><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> and the source term <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> for some <span><math><mi>p</mi><mo><</mo><mi>d</mi></math></span>. This provides an alternative and analytic proof of a result by N.V. Krylov (<em>Ann. Probab.</em>, 2021) concerning <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> drifts. The key step involves deriving a Gehring-type inequality for Green's functions by using the Fabes-Stroock approach (<em>Duke Math. J.</em>, 1984).</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103805"},"PeriodicalIF":2.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219418","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}
Pub Date : 2025-12-01Epub Date: 2025-06-13DOI: 10.1016/j.matpur.2025.103762
Yu-Chu Lin , Haitao Wang , Kung-Chien Wu
Consider the Boltzmann equation in the perturbation regime. Since the macroscopic quantities in the background global Maxwellian are obtained through measurements, there are typically some errors involved. This paper investigates the effect of background variations on the solution for a given initial perturbation. Our findings demonstrate that the solution changes continuously with variations in the background and provide a sharp time decay estimate of the associated errors. The proof relies on refined estimates for the linearized solution operator and a proper decomposition of the nonlinear solution.
{"title":"Stability of background perturbation for Boltzmann equation","authors":"Yu-Chu Lin , Haitao Wang , Kung-Chien Wu","doi":"10.1016/j.matpur.2025.103762","DOIUrl":"10.1016/j.matpur.2025.103762","url":null,"abstract":"<div><div>Consider the Boltzmann equation in the perturbation regime. Since the macroscopic quantities in the background global Maxwellian are obtained through measurements, there are typically some errors involved. This paper investigates the effect of background variations on the solution for a given initial perturbation. Our findings demonstrate that the solution changes continuously with variations in the background and provide a sharp time decay estimate of the associated errors. The proof relies on refined estimates for the linearized solution operator and a proper decomposition of the nonlinear solution.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103762"},"PeriodicalIF":2.1,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338507","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}
Pub Date : 2025-12-01Epub Date: 2025-06-13DOI: 10.1016/j.matpur.2025.103764
Verena Bögelein, Frank Duzaar, Naian Liao, Kristian Moring
We consider local weak solutions to the fractional p-Poisson equation of order s, i.e. . In the range and we prove Calderón & Zygmund type estimates at the gradient level. More precisely, we show for any that The qualitative result is accompanied by a local quantitative estimate.
{"title":"Gradient estimates for the fractional p-Poisson equation","authors":"Verena Bögelein, Frank Duzaar, Naian Liao, Kristian Moring","doi":"10.1016/j.matpur.2025.103764","DOIUrl":"10.1016/j.matpur.2025.103764","url":null,"abstract":"<div><div>We consider local weak solutions to the fractional <em>p</em>-Poisson equation of order <em>s</em>, i.e. <span><math><msup><mrow><mo>(</mo><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>f</mi></math></span>. In the range <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span> and <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mfrac><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mo>,</mo><mn>1</mn><mo>)</mo></math></span> we prove Calderón & Zygmund type estimates at the gradient level. More precisely, we show for any <span><math><mi>q</mi><mo>></mo><mn>1</mn></math></span> that<span><span><span><math><mi>f</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mfrac><mrow><mi>q</mi><mi>p</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msubsup><mspace></mspace><mo>⟹</mo><mspace></mspace><mi>∇</mi><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mi>q</mi><mi>p</mi></mrow></msubsup><mo>.</mo></math></span></span></span> The qualitative result is accompanied by a local quantitative estimate.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103764"},"PeriodicalIF":2.1,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338508","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}
Pub Date : 2025-12-01Epub Date: 2025-09-12DOI: 10.1016/j.matpur.2025.103797
Fabian Rupp , Christian Scharrer
We provide sharp sufficient criteria for an integral 2-varifold to be induced by a -conformal immersion of a smooth surface. Our approach is based on a fine analysis of the Hausdorff density for 2-varifolds with critical integrability of the mean curvature and a recent local regularity result by Bi–Zhou. In codimension one, there are only three possible density values below 2, each of which can be attained with equality in the Li–Yau inequality for the Willmore functional by the unit sphere, the double bubble, and the triple bubble. We show that below an optimal threshold for the Willmore energy, a varifold induced by a current without boundary is in fact a curvature varifold with a uniform bound on its second fundamental form. Consequently, the minimization of the Willmore functional in the class of curvature varifolds with prescribed even Euler characteristic provides smooth solutions for the Willmore problem. In particular, the “ambient” varifold approach and the “parametric” approach are equivalent for minimizing the Willmore energy.
{"title":"Global regularity of integral 2-varifolds with square integrable mean curvature","authors":"Fabian Rupp , Christian Scharrer","doi":"10.1016/j.matpur.2025.103797","DOIUrl":"10.1016/j.matpur.2025.103797","url":null,"abstract":"<div><div>We provide sharp sufficient criteria for an integral 2-varifold to be induced by a <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msup></math></span>-conformal immersion of a smooth surface. Our approach is based on a fine analysis of the Hausdorff density for 2-varifolds with critical integrability of the mean curvature and a recent local regularity result by Bi–Zhou. In codimension one, there are only three possible density values below 2, each of which can be attained with equality in the Li–Yau inequality for the Willmore functional by the unit sphere, the double bubble, and the triple bubble. We show that below an optimal threshold for the Willmore energy, a varifold induced by a current without boundary is in fact a curvature varifold with a uniform bound on its second fundamental form. Consequently, the minimization of the Willmore functional in the class of curvature varifolds with prescribed even Euler characteristic provides smooth solutions for the Willmore problem. In particular, the “ambient” varifold approach and the “parametric” approach are equivalent for minimizing the Willmore energy.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103797"},"PeriodicalIF":2.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145109261","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}
Pub Date : 2025-12-01Epub Date: 2025-09-12DOI: 10.1016/j.matpur.2025.103796
Simon Becker , Angeliki Menegaki , Jiming Yu
We consider chains of N harmonic oscillators in two dimensions coupled to a Langevin heat reservoir at fixed temperature, a classical model for heat conduction introduced by Lebowitz, Lieb, and Rieder (1967). We extend our previous results (Becker and Menegaki, 2021) significantly by providing a full spectral description of the full Fokker-Planck operator, also allowing for the presence of a constant external magnetic field for charged oscillators. We then study oscillator chains with additional next-to-nearest-neighbor interactions and find that the spectral gap undergoes a phase transition if the next-to-nearest-neighbor interactions are sufficiently strong and may even cease to exist for oscillator chains of finite length.
{"title":"Spectral analysis and phase transitions for long-range interactions in harmonic chains of oscillators","authors":"Simon Becker , Angeliki Menegaki , Jiming Yu","doi":"10.1016/j.matpur.2025.103796","DOIUrl":"10.1016/j.matpur.2025.103796","url":null,"abstract":"<div><div>We consider chains of <em>N</em> harmonic oscillators in two dimensions coupled to a Langevin heat reservoir at fixed temperature, a classical model for heat conduction introduced by Lebowitz, Lieb, and Rieder (1967). We extend our previous results (Becker and Menegaki, 2021) significantly by providing a full spectral description of the full Fokker-Planck operator, also allowing for the presence of a constant external magnetic field for charged oscillators. We then study oscillator chains with additional next-to-nearest-neighbor interactions and find that the spectral gap undergoes a phase transition if the next-to-nearest-neighbor interactions are sufficiently strong and may even cease to exist for oscillator chains of finite length.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103796"},"PeriodicalIF":2.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145109259","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}
Pub Date : 2025-11-01Epub Date: 2025-07-21DOI: 10.1016/j.matpur.2025.103772
Timo S. Hänninen , Emiel Lorist , Jaakko Sinko
<div><div>As our main result, we supply the missing characterization of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo><mo>→</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> boundedness of the commutator of a non-degenerate Calderón–Zygmund operator <em>T</em> and pointwise multiplication by <em>b</em> for exponents <span><math><mn>1</mn><mo><</mo><mi>q</mi><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo></math></span> and Muckenhoupt weights <span><math><mi>μ</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> and <span><math><mi>λ</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Namely, the commutator <span><math><mo>[</mo><mi>b</mi><mo>,</mo><mi>T</mi><mo>]</mo><mo>:</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo><mo>→</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> is bounded if and only if <em>b</em> satisfies the following new, cancellative condition:<span><span><span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mi>#</mi></mrow></msubsup><mi>b</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi><mi>q</mi><mo>/</mo><mo>(</mo><mi>p</mi><mo>−</mo><mi>q</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>ν</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mi>#</mi></mrow></msubsup><mi>b</mi></math></span> is the weighted sharp maximal function defined by<span><span><span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mi>#</mi></mrow></msubsup><mi>b</mi><mo>:</mo><mo>=</mo><munder><mi>sup</mi><mrow><mi>Q</mi></mrow></munder><mo></mo><mfrac><mrow><msub><mrow><mn>1</mn></mrow><mrow><mi>Q</mi></mrow></msub></mrow><mrow><mi>ν</mi><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></mfrac><munder><mo>∫</mo><mrow><mi>Q</mi></mrow></munder><mo>|</mo><mi>b</mi><mo>−</mo><msub><mrow><mo>〈</mo><mi>b</mi><mo>〉</mo></mrow><mrow><mi>Q</mi></mrow></msub><mo>|</mo><mspace></mspace><mi>d</mi><mi>x</mi></math></span></span></span> and <em>ν</em> is the Bloom weight defined by <span><math><msup><mrow><mi>ν</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msup><mo>:</mo><mo>=</mo><msup><mrow><mi>μ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>p</mi></mrow></msup><msup><mrow><mi>λ</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mi>q</mi></mrow></msup></math></span>.</div><div>In the unweighted case <span><math><mi>μ</mi><mo>=</mo><mi>λ</mi><mo>=</mo><mn>1</mn></math></span>, by a result of Hytönen the boundedness of the commutator <span><math><mo>[</mo><mi>b</mi><mo>,</mo><mi>T</mi><mo>]</mo></math></span> is, after factoring out constants, characterized by the boundedness
作为我们的主要结果，我们提供了对于指数1<；q<p<；∞和Muckenhoupt权μ∈Ap和λ∈Aq的非简并Calderón-Zygmund算子T的对易子的Lp（μ）→Lq（λ）有界性和点向乘b的缺失表征。即，换向子[b，T]:Lp（μ）→Lq（λ）有界当且仅当b满足以下新的可消条件：Mν#b∈Lpq/(p−q)(ν)，其中Mν#b是Mν#b定义的加权极大函数：=supQ (q)∫q |b−< b > q |dx， ν是ν1/p+1/q ':=μ1/pλ−1/q定义的Bloom权值。在μ=λ=1的未加权情况下，由Hytönen的结果可知，对易子[b，T]的有界性，在分解出常数后，表征为点向乘以b的有界性，即b∈Lpq/(p−q)为不可消去条件。我们提供了一个反例，表明在μ∈Ap和λ∈Aq的加权情况下，这种表征被打破。因此，引入新的消去条件是必要的。与对易子并行，我们也刻画了在缺失指数范围p≠q的并矢副积Πb的加权有界性。结合之前在互补指数范围内的结果，我们的结果完成了对所有指数p，q∈(1,∞)的对易子和副积的加权有界性的刻画。
{"title":"Weighted Lp → Lq-boundedness of commutators and paraproducts in the Bloom setting","authors":"Timo S. Hänninen , Emiel Lorist , Jaakko Sinko","doi":"10.1016/j.matpur.2025.103772","DOIUrl":"10.1016/j.matpur.2025.103772","url":null,"abstract":"<div><div>As our main result, we supply the missing characterization of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo><mo>→</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> boundedness of the commutator of a non-degenerate Calderón–Zygmund operator <em>T</em> and pointwise multiplication by <em>b</em> for exponents <span><math><mn>1</mn><mo><</mo><mi>q</mi><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo></math></span> and Muckenhoupt weights <span><math><mi>μ</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> and <span><math><mi>λ</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Namely, the commutator <span><math><mo>[</mo><mi>b</mi><mo>,</mo><mi>T</mi><mo>]</mo><mo>:</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo><mo>→</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> is bounded if and only if <em>b</em> satisfies the following new, cancellative condition:<span><span><span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mi>#</mi></mrow></msubsup><mi>b</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi><mi>q</mi><mo>/</mo><mo>(</mo><mi>p</mi><mo>−</mo><mi>q</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>ν</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mi>#</mi></mrow></msubsup><mi>b</mi></math></span> is the weighted sharp maximal function defined by<span><span><span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mi>#</mi></mrow></msubsup><mi>b</mi><mo>:</mo><mo>=</mo><munder><mi>sup</mi><mrow><mi>Q</mi></mrow></munder><mo></mo><mfrac><mrow><msub><mrow><mn>1</mn></mrow><mrow><mi>Q</mi></mrow></msub></mrow><mrow><mi>ν</mi><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></mfrac><munder><mo>∫</mo><mrow><mi>Q</mi></mrow></munder><mo>|</mo><mi>b</mi><mo>−</mo><msub><mrow><mo>〈</mo><mi>b</mi><mo>〉</mo></mrow><mrow><mi>Q</mi></mrow></msub><mo>|</mo><mspace></mspace><mi>d</mi><mi>x</mi></math></span></span></span> and <em>ν</em> is the Bloom weight defined by <span><math><msup><mrow><mi>ν</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msup><mo>:</mo><mo>=</mo><msup><mrow><mi>μ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>p</mi></mrow></msup><msup><mrow><mi>λ</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mi>q</mi></mrow></msup></math></span>.</div><div>In the unweighted case <span><math><mi>μ</mi><mo>=</mo><mi>λ</mi><mo>=</mo><mn>1</mn></math></span>, by a result of Hytönen the boundedness of the commutator <span><math><mo>[</mo><mi>b</mi><mo>,</mo><mi>T</mi><mo>]</mo></math></span> is, after factoring out constants, characterized by the boundedness","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103772"},"PeriodicalIF":2.3,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144860693","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}
Pub Date : 2025-11-01Epub Date: 2025-06-16DOI: 10.1016/j.matpur.2025.103752
Jiajun Tong , Yuming Paul Zhang
We investigate the general Porous Medium Equations with drift and source terms that model tumor growth. Incompressible limit of such models has been well-studied in the literature, where convergence of the density and pressure variables are established, while it remains unclear whether the free boundaries of the solutions exhibit convergence as well. In this paper, we provide an affirmative result by showing that the free boundaries converge in the Hausdorff distance in the incompressible limit. To achieve this, we quantify the relation between the free boundary motion and spatial average of the pressure, and establish a uniform-in-m strict expansion property of the pressure supports. As a corollary, we derive upper bounds for the Hausdorff dimensions of the free boundaries and show that the limiting free boundary has finite -dimensional Hausdorff measure.
{"title":"Convergence of free boundaries in the incompressible limit of tumor growth models","authors":"Jiajun Tong , Yuming Paul Zhang","doi":"10.1016/j.matpur.2025.103752","DOIUrl":"10.1016/j.matpur.2025.103752","url":null,"abstract":"<div><div>We investigate the general Porous Medium Equations with drift and source terms that model tumor growth. Incompressible limit of such models has been well-studied in the literature, where convergence of the density and pressure variables are established, while it remains unclear whether the free boundaries of the solutions exhibit convergence as well. In this paper, we provide an affirmative result by showing that the free boundaries converge in the Hausdorff distance in the incompressible limit. To achieve this, we quantify the relation between the free boundary motion and spatial average of the pressure, and establish a uniform-in-<em>m</em> strict expansion property of the pressure supports. As a corollary, we derive upper bounds for the Hausdorff dimensions of the free boundaries and show that the limiting free boundary has finite <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-dimensional Hausdorff measure.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103752"},"PeriodicalIF":2.1,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144320782","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}
In this work we analyse the small-time reachability properties of a nonlinear parabolic equation, by means of a bilinear control, posed on a torus of arbitrary dimension d. Under a saturation hypothesis on the control operators, we show the small-time approximate controllability between states sharing the same sign. Moreover, in the one-dimensional case , we combine this property with a local exact controllability result, and prove the small-time exact controllability of any positive states towards the ground state of the evolution operator.
Dans ce travail, nous analysons les propriétés d'accessibilité en temps court d'une équation parabolique non linéaire, à l'aide d'un contrôle bilinéaire, posée sur un tore de dimension arbitraire d. Sous une hypothèse de saturation sur les opérateurs de contrôle, nous montrons la contrôlabilité approchée en temps court entre les états qui ont le même signe. De plus, dans le cas unidimensionnel , nous combinons cette propriété avec un résultat de contrôlabilité locale exacte, et prouvons la contrôlabilité exacte en temps court de tout état positif vers l'état fondamental de l'opérateur d'évolution.
{"title":"On the small-time bilinear control of a nonlinear heat equation: Global approximate controllability and exact controllability to trajectories","authors":"Alessandro Duca , Eugenio Pozzoli , Cristina Urbani","doi":"10.1016/j.matpur.2025.103758","DOIUrl":"10.1016/j.matpur.2025.103758","url":null,"abstract":"<div><div>In this work we analyse the small-time reachability properties of a nonlinear parabolic equation, by means of a bilinear control, posed on a torus of arbitrary dimension <em>d</em>. Under a saturation hypothesis on the control operators, we show the small-time approximate controllability between states sharing the same sign. Moreover, in the one-dimensional case <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span>, we combine this property with a local exact controllability result, and prove the small-time exact controllability of any positive states towards the ground state of the evolution operator.</div><div>Dans ce travail, nous analysons les propriétés d'accessibilité en temps court d'une équation parabolique non linéaire, à l'aide d'un contrôle bilinéaire, posée sur un tore de dimension arbitraire <em>d</em>. Sous une hypothèse de saturation sur les opérateurs de contrôle, nous montrons la contrôlabilité approchée en temps court entre les états qui ont le même signe. De plus, dans le cas unidimensionnel <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span>, nous combinons cette propriété avec un résultat de contrôlabilité locale exacte, et prouvons la contrôlabilité exacte en temps court de tout état positif vers l'état fondamental de l'opérateur d'évolution.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103758"},"PeriodicalIF":2.1,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144320781","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}
Pub Date : 2025-11-01Epub Date: 2025-06-16DOI: 10.1016/j.matpur.2025.103756
Zhaosheng Feng , Junjie Zhang , Shenzhou Zheng
We consider a nonlinear elliptic equation of the form , where the principle part depends on the solution itself and the right-hand data μ is a signed Radon measure. The associated nonlinearity is assumed to satisfy the -BMO condition in x and the Lipschitz continuity condition in u, and its growth in Du is like the -Laplacian, while the boundary of underlying domain is assumed to be Reifenberg flat. We establish an optimal global Calderón-Zygmund type estimate in weighted Lorentz spaces for the gradients of very weak solutions to such a measure data problem. This is achieved by developing the perturbation method and modifying the weighted Vitali type covering argument.
{"title":"Weighted gradient estimates to nonlinear elliptic equations of p(x)-growth with measure data","authors":"Zhaosheng Feng , Junjie Zhang , Shenzhou Zheng","doi":"10.1016/j.matpur.2025.103756","DOIUrl":"10.1016/j.matpur.2025.103756","url":null,"abstract":"<div><div>We consider a nonlinear elliptic equation of the form <span><math><mo>−</mo><mtext>div</mtext><mi>A</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>)</mo><mo>=</mo><mi>μ</mi></math></span>, where the principle part depends on the solution itself and the right-hand data <em>μ</em> is a signed Radon measure. The associated nonlinearity is assumed to satisfy the <span><math><mo>(</mo><mi>δ</mi><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span>-BMO condition in <em>x</em> and the Lipschitz continuity condition in <em>u</em>, and its growth in <em>Du</em> is like the <span><math><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>-Laplacian, while the boundary of underlying domain is assumed to be Reifenberg flat. We establish an optimal global Calderón-Zygmund type estimate in weighted Lorentz spaces for the gradients of very weak solutions to such a measure data problem. This is achieved by developing the perturbation method and modifying the weighted Vitali type covering argument.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103756"},"PeriodicalIF":2.1,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144320783","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}
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