By Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci

This quantity comprises a number of invited lectures given on the foreign Workshop "Analysis, Partial Differential Equations and Applications", held on the Mathematical division of Sapienza college of Rome, at the social gathering 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 ebook goals at spreading the seminal rules of Maz'ya to a bigger viewers in schools of sciences and engineering. actually, all articles have been encouraged via prior works of Maz'ya in different frameworks, together with classical and modern difficulties attached with boundary and preliminary price difficulties for elliptic, hyperbolic and parabolic operators, Schr?dinger-type equations, mathematical thought of elasticity, strength idea, potential, singular indispensable operators, p-Laplacians, sensible research, and approximation concept. Maz'ya is writer of greater than 450 papers and 20 books. In his lengthy profession he bought many unbelievable and often pointed out leads to the speculation of harmonic potentials on non-smooth domain names, capability and means theories, areas of services with bounded version, greatest precept for higher-order elliptic equations, Sobolev multipliers, approximate approximations, and so on. the themes integrated during this quantity should be really worthwhile to all researchers who're drawn to attaining a deeper knowing of the massive services of Vladimir Maz'ya.

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**Example text**

Indeed, let us consider the degenerate diﬀusion process modelized by the Ito’s stochastic diﬀern 32 I. Capuzzo Dolcetta ential equation dXt = a(Xt ) dt + 2A(Xt ) dWt , X0 = x ∈ Ω , where Wt is a standard n-dimensional Brownian motion and a(Xt ), with |a| ≤ 1, is a feedback control acting on the trajectory Xt . 2) 0 where Ex is the conditional expectation with respect to the initial state x = X0 . 2), f, g are given continuous functions, q > 1 , cq = 1q 1 − 1q and λ ≥ 0 represents a discount rate.

Burago, V. Maz’ya, Potential Theory and Function Theory for Irregular Regions, Consultants Bureau, New York, 1969. [3] H. Federer, Geometric measure theory, Springer, Berlin, 1969. H. W. Rishel, An integral formula for total gradient variation, Arch. Math. 11 (1960), 218–222. [5] V. Maz’ya, Sobolev spaces, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985. [6] V. Maz’ya, Classes of regions and embedding theorems for functional spaces, Dokl. Akad. Nauk SSSR 133 (1960), 527–530.

As pointed out above this a very useful fact in the analysis of homogenization problems. 4. The result above can be extended to more general equations having a ﬁrst-order term of the form H(x, ξ) with H : Ω × Rn → R continuous and satisfying H(x, ξ) ≥ γ0 |ξ|p , |H(x, ξ) − H(x, η)| ≤ γ1 (|ξ| p−1 p > 1, + |η|p−1 )|ξ − η| , |H(x, ξ) − H(y, ξ)| ≤ γ2 (g1 (x) + g2 (x)|ξ|p ) |x − y| . 1 in the presence of degeneracies of the principal part. 38 I. Capuzzo Dolcetta 4. The Dirichlet problem It is well known from the fundamental works of Fichera [9] and Oleinik-Radkevic [17] that the degeneracy of the principal part of the equation is an obstruction to the solvability of the linear Dirichlet problem −Tr A(x)D2 u = f (x) in Ω , u =g on ∂Ω , in the classical sense for arbitrarily prescribed boundary data.