Ebook: Calculus of Variations and Partial Differential Equations: Topics on Geometrical Evolution Problems and Degree Theory

The link between Calculus of Variations and Partial Differential Equations has always been strong, because variational problems produce, via their Euler-Lagrange equation, a differential equation and, conversely, a differential equation can often be studied by variational methods. At the summer school in Pisa in September 1996, Luigi Ambrosio and Norman Dancer each gave a course on a classical topic (the geometric problem of evolution of a surface by mean curvature, and degree theory with applications to pde's resp.), in a self-contained presentation accessible to PhD students, bridging the gap between standard courses and advanced research on these topics. The resulting book is divided accordingly into 2 parts, and nicely illustrates the 2-way interaction of problems and methods. Each of the courses is augmented and complemented by additional short chapters by other authors describing current research problems and results.

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The link between Calculus of Variations and Partial Differential Equations has always been strong, because variational problems produce, via their Euler-Lagrange equation, a differential equation and, conversely, a differential equation can often be studied by variational methods. At the summer school in Pisa in September 1996, Luigi Ambrosio and Norman Dancer each gave a course on a classical topic (the geometric problem of evolution of a surface by mean curvature, and degree theory with applications to pde's resp.), in a self-contained presentation accessible to PhD students, bridging the gap between standard courses and advanced research on these topics. The resulting book is divided accordingly into 2 parts, and nicely illustrates the 2-way interaction of problems and methods. Each of the courses is augmented and complemented by additional short chapters by other authors describing current research problems and results.

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The link between Calculus of Variations and Partial Differential Equations has always been strong, because variational problems produce, via their Euler-Lagrange equation, a differential equation and, conversely, a differential equation can often be studied by variational methods. At the summer school in Pisa in September 1996, Luigi Ambrosio and Norman Dancer each gave a course on a classical topic (the geometric problem of evolution of a surface by mean curvature, and degree theory with applications to pde's resp.), in a self-contained presentation accessible to PhD students, bridging the gap between standard courses and advanced research on these topics. The resulting book is divided accordingly into 2 parts, and nicely illustrates the 2-way interaction of problems and methods. Each of the courses is augmented and complemented by additional short chapters by other authors describing current research problems and results.
Content:
Front Matter....Pages I-IX
Front Matter....Pages 1-1
Introduction to Part I....Pages 3-4
Geometric evolution problems, distance function and viscosity solutions....Pages 5-93
Variational models for phase transitions, an approach via ?-convergence....Pages 95-114
Some aspects of De Giorgi’s barriers for geometric evolutions....Pages 115-151
Partial Regularity for Minimizers of Free Discontinuity Problems with p-th Growth....Pages 153-169
Free discontinuity problems and their non-local approximation....Pages 171-180
Front Matter....Pages 181-181
Introduction to Part II....Pages 183-184
Degree theory on convex sets and applications to bifurcation....Pages 185-225
Nonlinear elliptic equations involving critical Sobolev exponents....Pages 227-241
On the existence and multiplicity of positive solutions for semilinear mixed and Neumann elliptic problems....Pages 243-258
Solitons and Relativistic Dynamics....Pages 259-283
An algebraic approach to nonstandard analysis....Pages 285-326
Back Matter....Pages 327-347

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The link between Calculus of Variations and Partial Differential Equations has always been strong, because variational problems produce, via their Euler-Lagrange equation, a differential equation and, conversely, a differential equation can often be studied by variational methods. At the summer school in Pisa in September 1996, Luigi Ambrosio and Norman Dancer each gave a course on a classical topic (the geometric problem of evolution of a surface by mean curvature, and degree theory with applications to pde's resp.), in a self-contained presentation accessible to PhD students, bridging the gap between standard courses and advanced research on these topics. The resulting book is divided accordingly into 2 parts, and nicely illustrates the 2-way interaction of problems and methods. Each of the courses is augmented and complemented by additional short chapters by other authors describing current research problems and results.
Content:
Front Matter....Pages I-IX
Front Matter....Pages 1-1
Introduction to Part I....Pages 3-4
Geometric evolution problems, distance function and viscosity solutions....Pages 5-93
Variational models for phase transitions, an approach via ?-convergence....Pages 95-114
Some aspects of De Giorgi’s barriers for geometric evolutions....Pages 115-151
Partial Regularity for Minimizers of Free Discontinuity Problems with p-th Growth....Pages 153-169
Free discontinuity problems and their non-local approximation....Pages 171-180
Front Matter....Pages 181-181
Introduction to Part II....Pages 183-184
Degree theory on convex sets and applications to bifurcation....Pages 185-225
Nonlinear elliptic equations involving critical Sobolev exponents....Pages 227-241
On the existence and multiplicity of positive solutions for semilinear mixed and Neumann elliptic problems....Pages 243-258
Solitons and Relativistic Dynamics....Pages 259-283
An algebraic approach to nonstandard analysis....Pages 285-326
Back Matter....Pages 327-347
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