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Table Of Integrals Series And Products

The marriage of analytic power to geometric intuition drives many of today's mathematical advances, yet books that build the connection from an elementary level remain scarce. This engaging introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. The theory of sets of finite perimeter provides a simple and effective framework. Topics covered include existence, regularity, analysis of singularities, characterization and symmetry results for minimizers in geometric variational problems, starting from the basics about Hausdorff measures in Euclidean spaces and ending with complete proofs of the regularity of area-minimizing hypersurfaces up to singular sets of codimension 8. Explanatory pictures, detailed proofs, exercises and remarks providing heuristic motivation and summarizing difficult arguments make this graduate-level textbook suitable for self-study and also a useful reference for researchers. Readers require only undergraduate analysis and basic measure theory.

/pdf/sets-of-finite-perimeter-and-geometric-variational-problems-20hqjijh88.pdf

Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory

Weakly Differentiable Functions

Handbook of Peridynamic Modeling

Progress in nonlinear differential equations and their applications, Vol. 80

In this paper, we study a new class of fractional partial differential equations which are obtained by minimizing variational problems in fractional Sobolev spaces. We introduce a notion of fractional gradient which has the potential to extend many classical results in the Sobolev spaces to the nonlocal and fractional setting in a natural way.

On a new class of fractional partial differential equations

http://www.math.cmu.edu/CNA/Publications/publications2011/003abs/11-CNA-003.pdf

Characterization of Sobolev and BV Spaces

In this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit of non-local functionals. One special case shows that rescalings of the fractional Laplacian of a function u ∈ S B V converge strictly to the singular portion of D u .

/pdf/on-formulae-decoupling-the-total-variation-of-bv-functions-3t99k7qru0.pdf

On formulae decoupling the total variation of BV functions

We prove the Boxing inequality: $$\mathcal{H}^{d-\alpha}_\infty(U) \leq C\alpha(1-\alpha)\int_U \int_{\mathbb{R}^{d} \setminus U} \frac{\mathrm{d}y \, \mathrm{d}z}{|y-z|^{\alpha+d}},$$ for every $\alpha \in (0,1)$ and every bounded open subset $U \subset \mathbb{R}^d$, where $\mathcal{H}^{d-\alpha}_\infty(U)$ is the Hausdorff content of $U$ of dimension $d -\alpha$ and the constant $C > 0$ depends only on $d$. We then show how this estimate implies a trace inequality in the fractional Sobolev space $W^{\alpha, 1}(\mathbb{R}^d)$ that includes Sobolev's $L^{\frac{d}{d - \alpha}}$ embedding, its Lorentz-space improvement, and Hardy's inequality. All these estimates are thus obtained with the appropriate asymptotics as $\alpha$ tends to $0$ and $1$, recovering in particular the classical inequalities of first order. Their counterparts in the full range $\alpha \in (0, d)$ are also investigated.

A Boxing Inequality for the Fractional Perimeter

We prove a family of Sobolev inequalities of the form $$ 
\Vert u \Vert_{L^{\frac{n}{n-1}, 1} (\mathbb{R}^n,V)} 
\le 
\Vert A (D) u \Vert_{L^1 (\mathbb{R}^n,E)} $$ where $A (D) : C^\infty_c (\mathbb{R}^n, V) \to C^\infty_c (\mathbb{R}^n, E)$ is a vector first-order homogeneous linear differential operator with constant coefficients, $u$ is a vector field on $\mathbb{R}^n$ and $L^{\frac{n}{n - 1}, 1} (\mathbb{R}^{n})$ is a Lorentz space. These new inequalities imply in particular the extension of the classical Gagliardo-Nirenberg inequality to Lorentz spaces originally due to Alvino and a sharpening of an inequality in terms of the deformation operator by Strauss (Korn-Sobolev inequality) on the Lorentz scale. The proof relies on a nonorthogonal application of the Loomis--Whitney inequality and Gagliardo's lemma.

Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo’s lemma

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