By Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci
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Extra resources for Analysis, partial differential equations and applications: The V.Maz'ya anniversary
1, 2, 14]. 3. 1 can be found in  for the equation −Δu + |Du|p + λ u = f (x) , x ∈ Ω, and in  for more general equations. The proof devised in  is not based, however, on the Bernstein method but rather on the tools introduced by H. L. Lions  which do not require diﬀerentiation of the equation, allowing therefore some x dependent nonlinearities. Observe also that in this way an estimate not depending on u ∞ , but only on λ u ∞ can be obtained. As pointed out above this a very useful fact in the analysis of homogenization problems.
2), f, g are given continuous functions, q > 1 , cq = 1q 1 − 1q and λ ≥ 0 represents a discount rate. Two cases of interest for the applications are when τx = inf t ≥ 0 : Xt ∈ Rn \ Ω or (the exit time problem) τx = +∞ (the state-constrained problem). The optimal control problem associated to these data is to minimize, for each given initial position X0 = x, the functional J(x, a) with respect to feedback controls a belonging to a speciﬁed set Ax of admissible controls. 1) with q p = q−1 . 4) at any point x0 ∈ Ω and for all C function Ψ touching from below the graph of u at x0 (the supersolution condition).
The Kaluza–Klein extension is a procedure which breaks conformal invariance, as one would expect when introducing mass. 2 which establishes that in the special case with no dependence on x3 our mathematical model is equivalent to the Dirac equation. This special case is known in literature as the Dirac equation in dimension 1 + 2 and is in itself the subject of extensive research. This leaves us with the question what can be said about the general case, when there is dependence on all spacetime coordinates (x0 , x1 , x2 , x3 ).
Analysis, partial differential equations and applications: The V.Maz'ya anniversary by Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci
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