Jürgen Appell, Agnieszka Chlebowicz, Simon Reinwand, Beata Rzepka
We analyse the “interplay” between analytical properties of a real function on a metric space, on the one hand, and topological properties of its graph, on the other. In particular, we study functions with closed, compact, connected, pathwise connected, or locally connected graphs, and we give nine conditions on the graph which are equivalent to the continuity of a function. A main emphasis is put on examples and counterexamples which illustrate how significant our hypotheses are, and how far sufficient conditions are from being necessary.
{"title":"Can one recognize a function from its graph?","authors":"Jürgen Appell, Agnieszka Chlebowicz, Simon Reinwand, Beata Rzepka","doi":"10.4171/zaa/1730","DOIUrl":"https://doi.org/10.4171/zaa/1730","url":null,"abstract":"We analyse the “interplay” between analytical properties of a real function on a metric space, on the one hand, and topological properties of its graph, on the other. In particular, we study functions with closed, compact, connected, pathwise connected, or locally connected graphs, and we give nine conditions on the graph which are equivalent to the continuity of a function. A main emphasis is put on examples and counterexamples which illustrate how significant our hypotheses are, and how far sufficient conditions are from being necessary.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136279753","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}
Nikolaos S. Papageorgiou, Francesca Vetro, Patrick Winkert
In this paper, we deal with a double phase problem with variable exponent and a right-hand side consisting of a Carathéodory perturbation defined only locally and of a critical term. We stress that the presence of the critical term inhibits the possibility to apply results of the critical point theory to the corresponding energy functional. Instead, we use suitable cut-off functions and truncation techniques in order to work with a coercive functional. Then, using variational tools and an appropriate auxiliary coercive problem, we can produce a sequence of sign changing solutions to our main problem converging to $0$ in $L^{infty}$ and in the Musielakk–Orlicz Sobolev space.
{"title":"Sign changing solutions for critical double phase problems with variable exponent","authors":"Nikolaos S. Papageorgiou, Francesca Vetro, Patrick Winkert","doi":"10.4171/zaa/1731","DOIUrl":"https://doi.org/10.4171/zaa/1731","url":null,"abstract":"In this paper, we deal with a double phase problem with variable exponent and a right-hand side consisting of a Carathéodory perturbation defined only locally and of a critical term. We stress that the presence of the critical term inhibits the possibility to apply results of the critical point theory to the corresponding energy functional. Instead, we use suitable cut-off functions and truncation techniques in order to work with a coercive functional. Then, using variational tools and an appropriate auxiliary coercive problem, we can produce a sequence of sign changing solutions to our main problem converging to $0$ in $L^{infty}$ and in the Musielakk–Orlicz Sobolev space.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136279756","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}
We are concerned with a problem described the deformation of a simply supported beam of the form $$ u^{(4)}(x)+c(x)u(x) + sum^p_{i=1}c_idelta(x-x_i)u(x) = lambda a(u(x)) + lambdasum^q_{j=1}a_j(u(x))delta(x-y_j), quad xin (0,1), $$ $$ u(0)=u(1)=u''(0)=u''(1)=0, $$ where $lambda$ is a positive parameter, $cin C([0, 1],mathbb{R})$, $c_i in mathbb{R}$, $a, a_jin C([0,infty),[0,infty))$, $i = 1, 2, ldots, p$, $j= 1, 2, ldots, q$, $p, q in mathbb{N}$. The Dirac delta impulses $delta = delta(x)$ are applied at given points $0 < x_1 < x_2
{"title":"Global structure of positive solutions for a fourth-order boundary value problem with singular data","authors":"Ruyun Ma, Zhongzi Zhao, Mantang Ma","doi":"10.4171/zaa/1729","DOIUrl":"https://doi.org/10.4171/zaa/1729","url":null,"abstract":"We are concerned with a problem described the deformation of a simply supported beam of the form $$ u^{(4)}(x)+c(x)u(x) + sum^p_{i=1}c_idelta(x-x_i)u(x) = lambda a(u(x)) + lambdasum^q_{j=1}a_j(u(x))delta(x-y_j), quad xin (0,1), $$ $$ u(0)=u(1)=u''(0)=u''(1)=0, $$ where $lambda$ is a positive parameter, $cin C([0, 1],mathbb{R})$, $c_i in mathbb{R}$, $a, a_jin C([0,infty),[0,infty))$, $i = 1, 2, ldots, p$, $j= 1, 2, ldots, q$, $p, q in mathbb{N}$. The Dirac delta impulses $delta = delta(x)$ are applied at given points $0 < x_1 < x_2 <cdots < x_p < 1$ and $0 < y_1 < y_2 < cdots < y_q < 1$. We investigate the global structure of positive solutions by the global bifurcation techniques.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136279755","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}
We establish Hardy–Sobolev and Hardy–Trudinger inequalities in weighted Orlicz spaces on $mathbb{R}^n$. As an application, we prove Hardy–Sobolev and Hardy–Trudinger inequalities in the framework of general double phase functionals given by $$ varphi_p(x,t) = varphi_1(t^p) + varphi_2((b(x)t)^p), quad xin mathbb{R}^n,, t ge 0, $$ where $p>1$, $varphi_1, varphi_2$ are positive convex functions on $(0,infty)$ and $b$ is a non-negative function on $[0,infty)$ which is Hölder continuous of order $theta in (0,1]$.
在$mathbb{R}^n$上建立了加权Orlicz空间中的Hardy-Sobolev不等式和Hardy-Trudinger不等式。作为应用，我们在$$ varphi_p(x,t) = varphi_1(t^p) + varphi_2((b(x)t)^p), quad xin mathbb{R}^n,, t ge 0, $$给出的一般双相泛函的框架内证明了Hardy-Sobolev和Hardy-Trudinger不等式，其中$p>1$、$varphi_1, varphi_2$是$(0,infty)$上的正凸函数，$b$是$[0,infty)$上的非负函数，它是Hölder阶$theta in (0,1]$连续的。
{"title":"Integrability for Hardy operators of double phase","authors":"Yoshihiro Mizuta, Tetsu Shimomura","doi":"10.4171/zaa/1732","DOIUrl":"https://doi.org/10.4171/zaa/1732","url":null,"abstract":"We establish Hardy–Sobolev and Hardy–Trudinger inequalities in weighted Orlicz spaces on $mathbb{R}^n$. As an application, we prove Hardy–Sobolev and Hardy–Trudinger inequalities in the framework of general double phase functionals given by $$ varphi_p(x,t) = varphi_1(t^p) + varphi_2((b(x)t)^p), quad xin mathbb{R}^n,, t ge 0, $$ where $p>1$, $varphi_1, varphi_2$ are positive convex functions on $(0,infty)$ and $b$ is a non-negative function on $[0,infty)$ which is Hölder continuous of order $theta in (0,1]$.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136279750","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}
We show that the embedding $dot{mathrm{BV}}(mathbb{R}^N)hookrightarrow L^{1^ast,q}(mathbb{R}^N)$, $q>1$ is cocompact with respect to the group and the profile decomposition for $dot{mathrm{BV}}(mathbb{R}^N)$. This paper extends the cocompactness and profile decomposition for the critical space $L^{1^ast}(mathbb{R}^N)$ to Lorentz spaces $L^{1^ast,q}(mathbb{R}^N)$, $q>1$. A~counterexample for $dot{mathrm{BV}}(mathbb{R}^N)hookrightarrow L^{1^ast,1}(mathbb{R}^N)$ not cocompact is given in the last section.
{"title":"Cocompact embedding theorem for functions of bounded variation into Lorentz spaces","authors":"Lin Zhao","doi":"10.4171/zaa/1728","DOIUrl":"https://doi.org/10.4171/zaa/1728","url":null,"abstract":"We show that the embedding $dot{mathrm{BV}}(mathbb{R}^N)hookrightarrow L^{1^ast,q}(mathbb{R}^N)$, $q>1$ is cocompact with respect to the group and the profile decomposition for $dot{mathrm{BV}}(mathbb{R}^N)$. This paper extends the cocompactness and profile decomposition for the critical space $L^{1^ast}(mathbb{R}^N)$ to Lorentz spaces $L^{1^ast,q}(mathbb{R}^N)$, $q>1$. A~counterexample for $dot{mathrm{BV}}(mathbb{R}^N)hookrightarrow L^{1^ast,1}(mathbb{R}^N)$ not cocompact is given in the last section.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690095","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}
The purpose of this paper is to provide a new proof of real interpolations for martingale Hardy–Orlicz–Lorentz spaces. Different from the usual proof, our approach does not rely on any atomic decompositions. Instead, we establish the result via new Holmstedt’s formulae in martingale Hardy–Orlicz type spaces.
{"title":"Real interpolations for martingale Hardy–Orlicz–Lorentz spaces","authors":"Wenfei Fan, Yanmeng Li, Lian Wu","doi":"10.4171/zaa/1726","DOIUrl":"https://doi.org/10.4171/zaa/1726","url":null,"abstract":"The purpose of this paper is to provide a new proof of real interpolations for martingale Hardy–Orlicz–Lorentz spaces. Different from the usual proof, our approach does not rely on any atomic decompositions. Instead, we establish the result via new Holmstedt’s formulae in martingale Hardy–Orlicz type spaces.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690252","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}
Modeling of multi-layered cell cultures (MCC) assume flat geometry in a 3D tumor growth model.Some results available in the time-dependent framework assume a periodic nutrient supply $Phi(t)$. Our aim is to generalize such considerations. Namely, we show that a stable oscillating tumor growth arises in the long term, whenever the mean value of the almost periodic nutrient supply $overline{Phi}$ surpasses certain threshold level, $overline{Phi}>tilde{sigma}$. Meanwhile, a vanishing tumor is predicted in the complementary case, $overline{Phi}leq tilde{sigma}$.
{"title":"Multi-layered tumor cell cultures with almost periodic nutrient supply","authors":"Homero G. Díaz-Marín","doi":"10.4171/zaa/1727","DOIUrl":"https://doi.org/10.4171/zaa/1727","url":null,"abstract":"Modeling of multi-layered cell cultures (MCC) assume flat geometry in a 3D tumor growth model.Some results available in the time-dependent framework assume a periodic nutrient supply $Phi(t)$. Our aim is to generalize such considerations. Namely, we show that a stable oscillating tumor growth arises in the long term, whenever the mean value of the almost periodic nutrient supply $overline{Phi}$ surpasses certain threshold level, $overline{Phi}>tilde{sigma}$. Meanwhile, a vanishing tumor is predicted in the complementary case, $overline{Phi}leq tilde{sigma}$.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135734885","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":"Global existence, asymptotic stability and blow up of solutions for a nonlinear viscoelastic plate equation involving (p(x); q(x))-Laplacian operator","authors":"M. Shahrouzi, J. Ferreira, F. Tahamtani","doi":"10.4171/zaa/1722","DOIUrl":"https://doi.org/10.4171/zaa/1722","url":null,"abstract":"","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45576763","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}
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