Pub Date : 2023-01-22DOI: 10.1512/iumj.2020.69.8412
K. Fouladgar, L. Simon
and, geometrically, A (u) represents the area functional for S(u); that is, A (u) is the (n+m−1)-dimensional Hausdorff measure H n+m−1(S(u)). This is clear because the integrand √ 1+|Du|2 um−1 for A (u) is the Jacobian of the map (x,ω) ∈Ω×Sm−1 7→ (x, u(x)ω)∈Ω×R, and this map is a local coordinate representation for the symmetric graph S(u). Since 1.1 1.1 expresses the fact that u is stationary with respect to A , we see that S(u) is stationary with respect to smooth symmetric deformations, and hence stationary with respect to all deformations by a well-known principle (see e.g. [Law72]). (The latter principle here is just the natural generalization of the fact that if a smooth hypersurface Σ is rotationally symmetric about an axis and if Σ is stationary with respect to smooth rotationally symmetric compactly supported perturbations, then Σ is minimal—i.e. stationary with respect to all smooth compactly supported perturbations whether symmetric or not.) Thus the smooth submanifold S(u) is stationary as a multiplicity 1 varifold in Ω× (R {0}) and hence is a smooth minimal submanifold of Ω× (R {0}) as claimed.
{"title":"The symmetric minimal surface equation","authors":"K. Fouladgar, L. Simon","doi":"10.1512/iumj.2020.69.8412","DOIUrl":"https://doi.org/10.1512/iumj.2020.69.8412","url":null,"abstract":"and, geometrically, A (u) represents the area functional for S(u); that is, A (u) is the (n+m−1)-dimensional Hausdorff measure H n+m−1(S(u)). This is clear because the integrand √ 1+|Du|2 um−1 for A (u) is the Jacobian of the map (x,ω) ∈Ω×Sm−1 7→ (x, u(x)ω)∈Ω×R, and this map is a local coordinate representation for the symmetric graph S(u). Since 1.1 1.1 expresses the fact that u is stationary with respect to A , we see that S(u) is stationary with respect to smooth symmetric deformations, and hence stationary with respect to all deformations by a well-known principle (see e.g. [Law72]). (The latter principle here is just the natural generalization of the fact that if a smooth hypersurface Σ is rotationally symmetric about an axis and if Σ is stationary with respect to smooth rotationally symmetric compactly supported perturbations, then Σ is minimal—i.e. stationary with respect to all smooth compactly supported perturbations whether symmetric or not.) Thus the smooth submanifold S(u) is stationary as a multiplicity 1 varifold in Ω× (R {0}) and hence is a smooth minimal submanifold of Ω× (R {0}) as claimed.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1512/iumj.2020.69.8412","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41881344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1512/iumj.2023.72.9335
Nguyen-Bac Dang, G. Tiozzo
{"title":"A central limit theorem for the degree of a random product of Cremona transformations","authors":"Nguyen-Bac Dang, G. Tiozzo","doi":"10.1512/iumj.2023.72.9335","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9335","url":null,"abstract":"","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66765754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1512/iumj.2023.72.9275
M. Grossi, Peng Luo
{"title":"Critical points of positive solutions of nonlinear elliptic equations: multiplicity, location, and non-degeneracy","authors":"M. Grossi, Peng Luo","doi":"10.1512/iumj.2023.72.9275","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9275","url":null,"abstract":"","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66766032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1512/iumj.2023.72.9773
Marco Capolli, A. Pinamonti, Gareth Speight
{"title":"A C^k Lusin approximation theorem for real-valued functions on Carnot groups","authors":"Marco Capolli, A. Pinamonti, Gareth Speight","doi":"10.1512/iumj.2023.72.9773","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9773","url":null,"abstract":"","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66766272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1512/iumj.2023.72.9769
Zongyuan Li, W. Wang
{"title":"Quantitative unique continuation for Robin boundary value problems on C^{1,1} domains","authors":"Zongyuan Li, W. Wang","doi":"10.1512/iumj.2023.72.9769","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9769","url":null,"abstract":"","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66766662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1512/iumj.2023.72.9137
Slawomir Dinew
We study a Monge-Ampere type equation in the class of $p$-plurisubharmonic functions and establish first and second order interior estimates. As an application of these we show that $p$-plurisubharmonic functions with constant operator and quadratic growth must be quadratic polynomials.
{"title":"Interior estimates for p-plurisubharmonic functions","authors":"Slawomir Dinew","doi":"10.1512/iumj.2023.72.9137","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9137","url":null,"abstract":"We study a Monge-Ampere type equation in the class of $p$-plurisubharmonic functions and establish first and second order interior estimates. As an application of these we show that $p$-plurisubharmonic functions with constant operator and quadratic growth must be quadratic polynomials.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135181154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1512/iumj.2023.72.9409
Ayman Rimah Said
We study the paralinearised weakly dispersive Burgers type equation: ∂tu+ ∂x[Tuu]− T ∂xu 2 u+ ∂x |D| α−1 u = 0, α ∈]1, 2[, which contains the main non linear ”worst interaction” terms, i.e low-high interaction terms, of the usual weakly dispersive Burgers type equation: ∂tu+ u∂xu+ ∂x |D| α−1 u = 0, α ∈]1, 2[, with u0 ∈ H (D), where D = T or R. Through a paradifferential complex Cole-Hopf type gauge transform we introduced in [38], we prove a new a priori estimate in H(D) under the control of ∥
研究了paralinearised弱色散汉堡类型方程:∂涂+∂x [Tuu]−T∂徐2 D u +∂x | |α−1 u = 0,α∈]1、2(,其中包含的主要非线性“最差互动”条款,即低交互方面,常见的弱色散汉堡类型方程:∂涂+ u∂徐D +∂x | |α−1 u = 0,α∈]1、2 (,,)uoh∈H (D), D = T或r .通过paradifferential复杂Cole-Hopf类型测量[38]中介绍了变换,我们证明一个新的先验估计在H (D)的控制下∥
{"title":"On the Cauchy problem for dispersive Burgers type equations","authors":"Ayman Rimah Said","doi":"10.1512/iumj.2023.72.9409","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9409","url":null,"abstract":"We study the paralinearised weakly dispersive Burgers type equation: ∂tu+ ∂x[Tuu]− T ∂xu 2 u+ ∂x |D| α−1 u = 0, α ∈]1, 2[, which contains the main non linear ”worst interaction” terms, i.e low-high interaction terms, of the usual weakly dispersive Burgers type equation: ∂tu+ u∂xu+ ∂x |D| α−1 u = 0, α ∈]1, 2[, with u0 ∈ H (D), where D = T or R. Through a paradifferential complex Cole-Hopf type gauge transform we introduced in [38], we prove a new a priori estimate in H(D) under the control of ∥","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66766254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1512/iumj.2023.72.9382
Guowei Dai
{"title":"Answer to some open problems and global bifurcation for an overdetermined problem","authors":"Guowei Dai","doi":"10.1512/iumj.2023.72.9382","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9382","url":null,"abstract":"","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135502959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1512/iumj.2023.72.9407
M. Šilhavý
{"title":"The Gauss-Green theorem for bounded vectorfields with divergence measure on sets of finite perimeter","authors":"M. Šilhavý","doi":"10.1512/iumj.2023.72.9407","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9407","url":null,"abstract":"","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66766197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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