Existence theorems for Lagrange problems in Sobolev spaces

1970 ◽  
pp. 39-49 ◽  
Author(s):  
Lamberto Cesari
Keyword(s):  
Sobolev Spaces ◽  
Existence Theorems ◽  
Lagrange Problems

An existence theorem for non-homogeneous differential inclusions in Sobolev spaces

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Jean-Philippe Mandallena ◽  
Mikhail Sychev
Keyword(s):  
Existence Theorem ◽  
Sobolev Spaces ◽  
Differential Inclusions ◽  
Existence Theorems ◽  
Lipschitz Functions ◽  
Higher Regularity ◽  
Funct Anal

Abstract In the present paper, we establish an existence theorem for non-homogeneous differential inclusions in Sobolev spaces. This theorem extends the results of Müller and Sychev [S. Müller and M. A. Sychev, Optimal existence theorems for nonhomogeneous differential inclusions, J. Funct. Anal. 181 2001, 2, 447–475; M. A. Sychev, Comparing various methods of resolving differential inclusions, J. Convex Anal. 18 2011, 4, 1025–1045] obtained in the setting of Lipschitz functions. We also show that solutions can be selected with the property of higher regularity.

Existence Theorems for a Class of Edge-Degenerate Elliptic Equations on Singular Manifolds

2015 ◽  
Vol 58 (2) ◽  
pp. 355-377
Author(s):  
Haining Fan
Keyword(s):  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Mountain Pass Theorem ◽  
Existence Theorems ◽  
Nehari Manifold ◽  
Mountain Pass ◽  
Degenerate Elliptic Equations ◽  
Degenerate Elliptic ◽  
The Nehari Manifold ◽  
Singular Manifolds

AbstractIn this paper we establish the Nehari manifold on edge Sobolev spaces and study some of their properties. Furthermore, we use these results and the mountain pass theorem to get non-negative solutions of a class of edge-degenerate elliptic equations on singular manifolds under different conditions.

Teichmuller Geometry

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Uniqueness Theorem ◽  
Existence Theorems ◽  
Quadratic Differentials ◽  
Complex Structures ◽  
Metric Geometry ◽  
Quasiconformal Maps ◽  
Hyperbolic Structures ◽  
Teichmüller Metric ◽  
Uniqueness And Existence ◽  
Teichmüller Mapping

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

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