Download Analysis, Partial Differential Equations and Applications: by Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci PDF

By Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci

This quantity contains numerous invited lectures given on the foreign Workshop "Analysis, Partial Differential Equations and Applications", held on the Mathematical division of Sapienza collage 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 colleges of sciences and engineering. in truth, all articles have been encouraged through earlier works of Maz'ya in numerous 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 idea of elasticity, capability conception, skill, singular critical operators, p-Laplacians, sensible research, and approximation thought. Maz'ya is writer of greater than 450 papers and 20 books. In his lengthy occupation he bought many astounding and regularly pointed out leads to the speculation of harmonic potentials on non-smooth domain names, power and capability theories, areas of capabilities 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 fairly helpful to all researchers who're drawn to reaching a deeper figuring out of the big services of Vladimir Maz'ya.

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Additional info for Analysis, Partial Differential Equations and Applications: The Vladimir Maz'ya Anniversary Volume (Operator Theory: Advances and Applications)

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1) for the covariant derivative of a spinor field is quite tricky). There is also a logical problem with the Dirac equation in that distinguishing the electron from the positron forces one to resort to the concept of negative energy. Finally, the electromagnetic field is incorporated into the Dirac equation by means of a formal substitution which does not admit a simple geometric interpretation. The purpose of this paper is to formulate an alternative mathematical model for the electron and positron, a model which is geometrically much simpler.

By ∇ we denote the covariant derivative with respect to the Levi-Civita connection. 1) 4 Dirac Equation 17 β respectively, where Γβ αγ = αγ := 12 g βδ (∂α gγδ +∂γ gαδ −∂δ gαγ ) are the Christoffel symbols and σβ are Pauli matrices. We identify differential forms with covariant antisymmetric tensors. Given a pair of real covariant antisymmetric tensors P and Q of rank r we define their dot 1 product as P · Q := r! βr g α1 β1 . . g αr βr . We also define P 2 := P · P . 3. The Dirac equation The following system of linear partial differential equations on M is known as the Dirac equation: ˙ σ αab (i∇ + A)α ηb˙ = mξ a , σ α ab˙ (i∇ + A)α ξ a = mηb˙ .

3. Lipschitz estimates In the paper [12] the authors consider C 2 solutions of the model equation −Δu + |Du|p + λu = f (x) , x ∈ Ω . 2) for some C > 0 depending on the data f and λ . Once it is known that any solution is locally Lipschitz, then the following precise estimate 1 p−1 |Du(x)| ≤ C d∂Ω (x) can be derived by using a scaling argument relying on the homogeneity of the equation. 1) and, for fixed x0 ∈ Ω, let r = d∂Ω (x0 ) . The scaled function 1 x ∈ B1 (0) , v(x) = α u(x0 + rx) r with α = p−2 p−1 satisfies −Δv + |Dv|p + λ r2 v = r2−α f (x0 + rx) x ∈ B1 (0) .

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