{"title":"Regolarità delle soluzioni di viscosità dell'equazione iconale","authors":"P. Albano","doi":"10.6092/ISSN.2240-2829/2255","DOIUrl":"https://doi.org/10.6092/ISSN.2240-2829/2255","url":null,"abstract":"We study the regularity of a viscosity solution of equations of eikonal type in two dierent framework: we consider either the non-degenerate eikonal equation under low regularity assumptions on the data or a possibly degenerate eikonal equation with data having analytic regularity. Furthermore, we describe the structure of the singular set.","PeriodicalId":41199,"journal":{"name":"Bruno Pini Mathematical Analysis Seminar","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2010-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71262265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Vector valued Fourier multipliers and applications","authors":"D. Guidetti","doi":"10.6092/ISSN.2240-2829/2231","DOIUrl":"https://doi.org/10.6092/ISSN.2240-2829/2231","url":null,"abstract":"In questo seminario sono illustrati alcuni recenti sviluppi della teoria dei moltiplicatori di Fourier negli spazi L^p a valori in spazi di Banach. Seguono alcune applicazioni a problemi al contorno di tipo ellittico e a problemi misti di tipo parabolico.","PeriodicalId":41199,"journal":{"name":"Bruno Pini Mathematical Analysis Seminar","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2010-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71262065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Local Monotonicity and Isoperimetric Inequality on Hypersurfaces in Carnot groups","authors":"F. Montefalcone","doi":"10.6092/ISSN.2240-2829/2254","DOIUrl":"https://doi.org/10.6092/ISSN.2240-2829/2254","url":null,"abstract":"Let G be a k-step Carnot group of homogeneous dimension Q. Later on we shall present some of the results recently obtained in [32] and, in particular, an intrinsic isoperimetric inequality for a C2-smooth compact hypersurface S with boundary @S. We stress that S and @S are endowed with the homogeneous measures n????1 H and n????2 H , respectively, which are actually equivalent to the intrinsic (Q - 1)-dimensional and (Q - 2)-dimensional Hausdor measures with respect to a given homogeneous metric % on G. This result generalizes a classical inequality, involving the mean curvature of the hypersurface, proven by Michael and Simon [29] and Allard [1], independently. One may also deduce some related Sobolev-type inequalities. The strategy of the proof is inspired by the classical one and will be discussed at the rst section. After reminding some preliminary notions about Carnot groups, we shall begin by proving a linear isoperimetric inequality. The second step is a local monotonicity formula. Then we may achieve the proof by a covering argument. We stress however that there are many dierences, due to our non-Euclidean setting. Some of the tools developed ad hoc are, in order, a blow-up\" theorem, which holds true also for characteristic points, and a smooth Coarea Formula for the HS-gradient. Other tools are the horizontal integration by parts formula and the 1st variation formula for the H-perimeter n????1 H already developed in [30, 31] and then generalized to hypersurfaces having non-empty characteristic set in [32]. These results can be useful in the study of minimal and constant horizontal mean curvature hypersurfaces in Carnot groups.","PeriodicalId":41199,"journal":{"name":"Bruno Pini Mathematical Analysis Seminar","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2010-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71262182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sull’equazione det Du = f senza ipotesi di segno","authors":"G. Cupini","doi":"10.6092/ISSN.2240-2829/2258","DOIUrl":"https://doi.org/10.6092/ISSN.2240-2829/2258","url":null,"abstract":"We consider the nonlinear problem det?u (x) = f (x) x ? u (x) = x x ? ? where k ? 1 is an integer, is a bounded smooth domain in Rn and f ? Ck ???? satisfies Z f (x) dx = meas . The positivity of f is a standard assumption in the literature. In a recent joint paper with B.Dacorogna and O.Kneuss (EPFL) we prove the existence of a solution u ? Ck ???? ;Rn with no assumptions on the sign of f. Here we state this theorem together with some related results and we outline the main features of the problem.","PeriodicalId":41199,"journal":{"name":"Bruno Pini Mathematical Analysis Seminar","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2010-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71262536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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