Proceedings of the Royal Society of Edinburgh Section A-Mathematics最新文献

{"title":"On a critical time-harmonic Maxwell equation in nonlocal media","authors":"Minbo Yang, Weiwei Ye, Shuijin Zhang","doi":"10.1017/prm.2024.11","DOIUrl":"https://doi.org/10.1017/prm.2024.11","url":null,"abstract":"In this paper, we study the existence of solutions for a critical time–harmonic Maxwell equation in nonlocal media <jats:disp-formula> <jats:alternatives> <jats:tex-math>[ begin{cases} nablatimes(nablatimes u)+lambda u=left(I_{alpha}ast|u|^{2^{{ast}}_{alpha}}right)|u|^{2^{{ast}}_{alpha}-2}u &amp; mathrm{in} Omega, nutimes u=0 &amp; mathrm{on} partialOmega, end{cases} ]</jats:tex-math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0308210524000118_eqnU1.png\" /> </jats:alternatives> </jats:disp-formula>where <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Omega subset mathbb {R}^{3}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000118_inline1.png\" /> </jats:alternatives> </jats:inline-formula> is a bounded domain, either convex or with <jats:inline-formula> <jats:alternatives> <jats:tex-math>$mathcal {C}^{1,1}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000118_inline2.png\" /> </jats:alternatives> </jats:inline-formula> boundary, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$nu$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000118_inline3.png\" /> </jats:alternatives> </jats:inline-formula> is the exterior normal, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$lambda &lt;0$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000118_inline4.png\" /> </jats:alternatives> </jats:inline-formula> is a real parameter, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$2^{ast }_{alpha }=3+alpha$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000118_inline5.png\" /> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:tex-math>$0&lt;alpha &lt;3$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000118_inline6.png\" /> </jats:alternatives> </jats:inline-formula> is the upper critical exponent due to the Hardy–Littlewood–Sobolev inequality. By introducing some suitable Coulomb spaces involving curl operator <jats:inline-formula> <jats:alternatives> <jats:tex-math>$W^{alpha,2^{ast }_{alpha }}_{0}(mathrm {curl};Omega )$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000118_inline7.png\" /> </jats:alternatives> </jats:inline-formula>, we are able to obtain the ground state solutions of the curl–curl equation via the method of constraining Nehari–Pankov manifold. Correspondingly, some sharp constants of the Sobolev-like inequalities with curl operator are obtained by a nonlocal version of the con","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"6 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140011185","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":"Multiplicity of positive solutions for a class of nonhomogeneous elliptic equations in the hyperbolic space","authors":"Debdip Ganguly, Diksha Gupta, K. Sreenadh","doi":"10.1017/prm.2024.18","DOIUrl":"https://doi.org/10.1017/prm.2024.18","url":null,"abstract":"The paper is concerned with positive solutions to problems of the type <jats:disp-formula> <jats:alternatives> <jats:tex-math>[ -Delta_{mathbb{B}^{N}} u - lambda u = a(x) |u|^{p-1};u + f text{ in }mathbb{B}^{N}, quad u in H^{1}{(mathbb{B}^{N})}, ]</jats:tex-math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0308210524000180_eqnU1.png\" /> </jats:alternatives> </jats:disp-formula>where <jats:inline-formula> <jats:alternatives> <jats:tex-math>$mathbb {B}^N$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000180_inline1.png\" /> </jats:alternatives> </jats:inline-formula> denotes the hyperbolic space, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$1&lt; p&lt;2^*-1:=frac {N+2}{N-2}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000180_inline2.png\" /> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$;lambda &lt; frac {(N-1)^2}{4}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000180_inline3.png\" /> </jats:alternatives> </jats:inline-formula>, and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$f in H^{-1}(mathbb {B}^{N})$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000180_inline4.png\" /> </jats:alternatives> </jats:inline-formula> (<jats:inline-formula> <jats:alternatives> <jats:tex-math>$f not equiv 0$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000180_inline5.png\" /> </jats:alternatives> </jats:inline-formula>) is a non-negative functional. The potential <jats:inline-formula> <jats:alternatives> <jats:tex-math>$ain L^infty (mathbb {B}^N)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000180_inline6.png\" /> </jats:alternatives> </jats:inline-formula> is assumed to be strictly positive, such that <jats:inline-formula> <jats:alternatives> <jats:tex-math>$lim _{d(x, 0) rightarrow infty } a(x) rightarrow 1,$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000180_inline7.png\" /> </jats:alternatives> </jats:inline-formula> where <jats:inline-formula> <jats:alternatives> <jats:tex-math>$d(x,, 0)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000180_inline8.png\" /> </jats:alternatives> </jats:inline-formula> denotes the geodesic distance. First, the existence of three positive solutions is proved under the assumption that <jats:inline-formula> <jats:alternatives> <jats:tex-math>$a(x) leq 1$</jats:tex-math> <jats:i","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"53 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140003467","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":"A collision result for both non-Newtonian and heat conducting Newtonian compressible fluids","authors":"Šárka Nečasová, Florian Oschmann","doi":"10.1017/prm.2024.5","DOIUrl":"https://doi.org/10.1017/prm.2024.5","url":null,"abstract":"<p>We generalize the known collision results for a solid in a 3D compressible Newtonian fluid to compressible non-Newtonian ones, and to Newtonian fluids with temperature-depending viscosities.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"23 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139969535","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":"Qualitative properties of solutions for system involving the fractional Laplacian","authors":"Ran Zhuo, Yingshu Lü","doi":"10.1017/prm.2024.10","DOIUrl":"https://doi.org/10.1017/prm.2024.10","url":null,"abstract":"<p>In this paper, we consider the following non-linear system involving the fractional Laplacian<span><span>0.1</span><span data-mathjax-type=\"texmath\"><span>begin{equation} left{begin{array}{@{}ll} (-Delta)^{s} u (x)= f(u,,v), (-Delta)^{s} v (x)= g(u,,v), end{array} right. end{equation}</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223140551915-0053:S0308210524000106:S0308210524000106_eqn1.png\"/></span>in two different types of domains, one is bounded, and the other is an infinite cylinder, where <span><span><span data-mathjax-type=\"texmath\"><span>$0&lt; s&lt;1$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223140551915-0053:S0308210524000106:S0308210524000106_inline1.png\"/></span></span>. We employ the direct sliding method for fractional Laplacian, different from the conventional extension and moving planes methods, to derive the monotonicity of solutions for (0.1) in <span><span><span data-mathjax-type=\"texmath\"><span>$x_n$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223140551915-0053:S0308210524000106:S0308210524000106_inline2.png\"/></span></span> variable. Meanwhile, we develop a new iteration method for systems in the proofs. Hopefully, the iteration method can also be applied to solve other problems.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"2 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139969777","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":"Nonlocal anisotropic interactions of Coulomb type","authors":"Maria Giovanna Mora","doi":"10.1017/prm.2024.19","DOIUrl":"https://doi.org/10.1017/prm.2024.19","url":null,"abstract":"<p>In this paper, we review some recent results on nonlocal interaction problems. The focus is on interaction kernels that are anisotropic variants of the classical Coulomb kernel. In other words, while preserving the same singularity at zero of the Coulomb kernel, they present preferred directions of interaction. For kernels of this kind and general confinement we will prove existence and uniqueness of minimizers of the corresponding energy. In the case of a quadratic confinement we will review a recent result by Carrillo and Shu about the explicit characterization of minimizers, and present a new proof, which has the advantage of being extendable to higher dimensions. In light of this result, we will re-examine some previous works motivated by applications to dislocation theory in materials science. Finally, we will discuss some related results and open questions.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"3 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139969770","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":"Common valuations of division polynomials","authors":"Bartosz Naskręcki, Matteo Verzobio","doi":"10.1017/prm.2024.7","DOIUrl":"https://doi.org/10.1017/prm.2024.7","url":null,"abstract":"<p>In this note, we prove a formula for the cancellation exponent <span><span><span data-mathjax-type=\"texmath\"><span>$k_{v,n}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline1.png\"/></span></span> between division polynomials <span><span><span data-mathjax-type=\"texmath\"><span>$psi _n$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline2.png\"/></span></span> and <span><span><span data-mathjax-type=\"texmath\"><span>$phi _n$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline3.png\"/></span></span> associated with a sequence <span><span><span data-mathjax-type=\"texmath\"><span>${nP}_{nin mathbb {N}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline4.png\"/></span></span> of points on an elliptic curve <span><span><span data-mathjax-type=\"texmath\"><span>$E$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline5.png\"/></span></span> defined over a discrete valuation field <span><span><span data-mathjax-type=\"texmath\"><span>$K$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline6.png\"/></span></span>. The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"14 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139969783","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 the Γ-convergence of the Allen–Cahn functional with boundary conditions","authors":"Dimitrios Gazoulis","doi":"10.1017/prm.2024.4","DOIUrl":"https://doi.org/10.1017/prm.2024.4","url":null,"abstract":"We study minimizers of the Allen–Cahn system. We consider the <jats:inline-formula> <jats:alternatives> <jats:tex-math>$varepsilon$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000040_inline2.png\" /> </jats:alternatives> </jats:inline-formula>-energy functional with Dirichlet values and we establish the <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Gamma$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000040_inline3.png\" /> </jats:alternatives> </jats:inline-formula>-limit. The minimizers of the limiting functional are closely related to minimizing partitions of the domain. Finally, utilizing that the triod and the straight line are the only minimal cones in the plane together with regularity results for minimal curves, we determine the precise structure of the minimizers of the limiting functional, and thus the limit of minimizers of the <jats:inline-formula> <jats:alternatives> <jats:tex-math>$varepsilon$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000040_inline4.png\" /> </jats:alternatives> </jats:inline-formula>-energy functional as <jats:inline-formula> <jats:alternatives> <jats:tex-math>$varepsilon rightarrow 0$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000040_inline5.png\" /> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"151 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758112","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 the Sobolev stability threshold for shear flows near Couette in 2D MHD equations","authors":"Ting Chen, Ruizhao Zi","doi":"10.1017/prm.2024.6","DOIUrl":"https://doi.org/10.1017/prm.2024.6","url":null,"abstract":"In this work, we study the Sobolev stability of shear flows near Couette in the 2D incompressible magnetohydrodynamics (MHD) equations with background magnetic field <jats:inline-formula> <jats:alternatives> <jats:tex-math>$(alpha,0 )^top$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000064_inline1.png\" /> </jats:alternatives> </jats:inline-formula> on <jats:inline-formula> <jats:alternatives> <jats:tex-math>$mathbb {T}times mathbb {R}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000064_inline2.png\" /> </jats:alternatives> </jats:inline-formula>. More precisely, for sufficiently large <jats:inline-formula> <jats:alternatives> <jats:tex-math>$alpha$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000064_inline3.png\" /> </jats:alternatives> </jats:inline-formula>, we show that when the initial datum of the shear flow satisfies <jats:inline-formula> <jats:alternatives> <jats:tex-math>$left | U(y)-yright |_{H^{N+6}}ll 1$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000064_inline4.png\" /> </jats:alternatives> </jats:inline-formula>, with <jats:inline-formula> <jats:alternatives> <jats:tex-math>$N&gt;1$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000064_inline5.png\" /> </jats:alternatives> </jats:inline-formula>, and the initial perturbations <jats:inline-formula> <jats:alternatives> <jats:tex-math>${u}_{mathrm {in}}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000064_inline6.png\" /> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>${b}_{mathrm {in}}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000064_inline7.png\" /> </jats:alternatives> </jats:inline-formula> satisfy <jats:inline-formula> <jats:alternatives> <jats:tex-math>$left | ( {u}_{mathrm {in}},{b}_{mathrm {in}}) right | _{H^{N+1}}=epsilon ll nu ^{frac 56+tilde delta }$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000064_inline8.png\" /> </jats:alternatives> </jats:inline-formula> for any fixed <jats:inline-formula> <jats:alternatives> <jats:tex-math>$tilde delta &gt;0$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000064_inline9.png\" /> </jats:alternatives> </jats:inline-formula>, then the solution of the 2D MHD equations remains <jats:inline-formula> <jats:alternatives> <jats:tex-math>$nu ^{-(frac {1}{3}+frac {tilde delta }{2})}epsilon$</jats:tex-math> <jats:i","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"10 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758157","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":"Remarks on a formula of Ramanujan","authors":"Andrés Chirre, Steven M. Gonek","doi":"10.1017/prm.2023.136","DOIUrl":"https://doi.org/10.1017/prm.2023.136","url":null,"abstract":"Assuming an averaged form of Mertens’ conjecture and that the ordinates of the non-trivial zeros of the Riemann zeta function are linearly independent over the rationals, we analyse the finer structure of the terms in a well-known formula of Ramanujan.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"6 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758154","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 the existence of a nodal solution for p-Laplacian equations depending on the gradient","authors":"F. Faraci, D. Puglisi","doi":"10.1017/prm.2023.135","DOIUrl":"https://doi.org/10.1017/prm.2023.135","url":null,"abstract":"<p>In the present paper we deal with a quasi-linear elliptic equation depending on a sublinear nonlinearity involving the gradient. We prove the existence of a nontrivial nodal solution employing the theory of invariant sets of descending flow together with sub-supersolution techniques, gradient regularity arguments, strong comparison principle for the <span><span><span data-mathjax-type=\"texmath\"><span>$p$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130151231725-0751:S030821052300135X:S030821052300135X_inline2.png\"/></span></span>-Laplace operator. The same conclusion is obtained for an eigenvalue problem under a different set of assumptions.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"103 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647806","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}