Lectures related to Sobolev inequality

Sobolev Spaces and Continuous Embeddings

Covers Sobolev spaces, continuous embeddings, weak convergence, and Poincare inequalities.

Exact Fields, Direct Methods, Sobolev Spaces

Covers Hilbert's theorem, direct methods, and Sobolev spaces, including classical and modern methods.

Covers Sobolev spaces, Lax-Milgram theorem, embeddings, and Lipschitz boundaries in PDEs.

Explores integration by parts in Sobolev spaces and the Poincaré inequality in

W^{1,p}_0

with applications in PDEs.

Embedding Theorems in Sobolev Spaces

Explores embedding theorems in Sobolev spaces, including continuous and compact embedding, weak convergence, and Poincaré inequality.

Compact Embedding: Theorem and Sobolev Inequalities

Covers the concept of compact embedding in Banach spaces and Sobolev inequalities.

Extension Theorems: Sobolev Spaces and Poincaré Inequality

Covers extension theorems in Sobolev spaces and Poincaré inequality, emphasizing the importance of understanding these concepts in partial differential equations.

Analysis IV: Le Spaces

Introduces Le spaces, measurable functions, Holder inequality, and LP space properties.

Embedding Theorems in Sobolev Spaces

Explores weak convergence, Poincaré inequality, and embedding theorems in Sobolev spaces.

Spectral Theory: Regularity and Embedding

Explores spectral theory, emphasizing regularity properties and embedding theorems in the context of Sobolev spaces and compactly supported functions.

Extension of Linear Transformations

Covers the extension of bounded linear transformations and the free propagator in L^2 spaces.

Approximation in Sobolev Spaces

Covers the approximation of functions in Sobolev spaces using smooth functions.

Embedding in Function Spaces

Explores embedding in function spaces, including elliptic equations, Hölder inequality, Lorentz space, and Hardy-Young inequality.

Sobolev Spaces in Higher Dimensions

Explores Sobolev spaces in higher dimensions, discussing derivatives, properties, and challenges with continuity.

Lp Spaces: Introduction

Introduces Lp spaces, covering norms, inequalities, and integrability of functions.

Properties of Weak Derivatives

Explores weak derivatives in Sobolev spaces, discussing their properties and uniqueness.

Measure Spaces: O-Finite and Probability Measures

Explores o-finite and finite measure spaces, probability measures, and inequalities, concluding with LP space completeness.

Density Results in Sobolev Spaces

Explores density results in Sobolev spaces, covering proofs and applications in various contexts.

Euler-Lagrange Equation

Covers the Euler-Lagrange equation in Sobolev spaces and discusses minimization, convexity, and weak forms.

Theoretical Study of Elliptic Partial Differential Equations

Covers the theoretical study of Elliptic Partial Differential Equations, including classical and weak solutions.