By Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci
This quantity comprises numerous invited lectures given on the overseas Workshop "Analysis, Partial Differential Equations and Applications", held on the Mathematical division of Sapienza college of Rome, at the party of the seventieth birthday of Vladimir G. Maz'ya, a well known mathematician and one of many major specialists within the box of natural and utilized research. The booklet goals at spreading the seminal rules of Maz'ya to a bigger viewers in colleges of sciences and engineering. in truth, all articles have been encouraged through prior works of Maz'ya in different frameworks, together with classical and modern difficulties attached with boundary and preliminary worth difficulties for elliptic, hyperbolic and parabolic operators, Schr?dinger-type equations, mathematical idea of elasticity, capability conception, potential, singular essential operators, p-Laplacians, practical research, and approximation idea. Maz'ya is writer of greater than 450 papers and 20 books. In his lengthy profession he got many miraculous and regularly pointed out leads to the speculation of harmonic potentials on non-smooth domain names, strength and capability theories, areas of services with bounded version, greatest precept for higher-order elliptic equations, Sobolev multipliers, approximate approximations, and so on. the subjects integrated during this quantity might be quite important to all researchers who're attracted to attaining a deeper knowing of the big services of Vladimir Maz'ya.
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Extra resources for Analysis, Partial Differential Equations and Applications: The Vladimir Maz'ya Anniversary Volume (Operator Theory: Advances and Applications)
The H¨olderianity up to ∂Ω can be obtained under smoothness assumptions on the boundary of Ω . 7. If u : B → R is a continuous function such that |x − y| |u(x) − u(y)| ≤ K + |x − y|α (d∂B (x) ∧ d∂B (y))1−α for some K > 0 and 0 < α < 1, then u can be extended up to B as a function satisfying |u(x) − u(y)| ≤ K |x − y|α for all x, y ∈ B where K ≥ K is a constant depending on α, K. We refer to  for the proof. 3. Lipschitz estimates In the paper  the authors consider C 2 solutions of the model equation −Δu + |Du|p + λu = f (x) , x ∈ Ω .
If moreover A is a real coeﬃcient operator, L is Lp -dissipative for any p.
Federer, Geometric measure theory, Springer, Berlin, 1969. H. W. Rishel, An integral formula for total gradient variation, Arch. Math. 11 (1960), 218–222.  V. Maz’ya, Sobolev spaces, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985.  V. Maz’ya, Classes of regions and embedding theorems for functional spaces, Dokl. Akad. Nauk SSSR 133 (1960), 527–530.  F. , San Diego, CA, 2000. Yuri Burago Saint-Petersburg branch of Steklov Math Institute Fontanka 27 St. ru Nikolay N.