Examinando por Autor "Porretta, Alessio"

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    Comparison principles for p-Laplace equations with lower order terms
    (Springer Nature, 2016-08-06) Leonori, Tommaso; Porretta, Alessio; Riey, Giuseppe
    We prove comparison principles for quasilinear elliptic equations whose simplest model is (Formula presented.), where Δpu=div(|Du|p-2Du) is the p-Laplace operator with p> 2 , λ≥ 0 , H(x, ξ) : Ω × RN→ R is a Carathéodory function and Ω ⊂ RN is a bounded domain, N≥ 2. We collect several comparison results for weak sub- and super-solutions under different setting of assumptions and with possibly different methods. A strong comparison result is also proved for more regular solutions.
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    Gradient Bounds for Elliptic Problems Singular at the Boundary
    (Springer Nature, 2011-07-05) Tommaso, Leonori; Porretta, Alessio
    Let Ω be a bounded smooth domain in RN, N ≧ 2, and let us denote by d(x) the distance function d(x, ∂Ω). We study a class of singular Hamilton-Jacobi equations, arising from stochastic control problems, whose simplest model is where f belongs to W 1,∞ loc (Ω) and is (possibly) singular at ∂Ω, C ε W1,∞ (Ω)(with no sign condition) and the field B ε W1,∞ (Ω)N has an outward direction and satisfies B · v ≧ α at ∂Ω (ν is the outward normal). Despite the singularity in the equation, we prove gradient bounds up to the boundary and the existence of a (globally) Lipschitz solution. We show that in some cases this is the unique bounded solution. We also discuss the stability of such estimates with respect to α, as α vanishes, obtaining Lipschitz solutions for first order problems with similar features. The main tool is a refined weighted version of the classical Bernstein method to get gradient bounds; the key role is played here by the orthogonal transport component of the Hamiltonian.
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    Large solutions and gradient bounds for quasilinear elliptic equations
    (Taylor and Francis Group, 2015-10-09) Leonori, Tommaso; Porretta, Alessio
    We consider the quasilinear degenerate elliptic equation λu - Δpu + H(x,Du)= 0 in Ω where Δp is the p-Laplace operator, p > 2, λ ≥0 and Ω is a smooth open bounded subset of ℝN (N ≥ 2). Under suitable structure conditions on the function H, we prove local and global gradient bounds for the solutions. We apply these estimates to study the solvability of the Dirichlet problem, and the existence, uniqueness and asymptotic behavior of maximal solutions blowing up at the boundary. The ergodic limit for those maximal solutions is also studied and the existence and uniqueness of a so-called additive eigenvalue is proved in this context.
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    On the comparison principle for unbounded solutions of elliptic equations with first order terms
    (Elsevier, 2018-01-15) Leonori, Tommaso; Porretta, Alessio
    We prove a comparison principle for unbounded weak sub/super solutions of the equation λu−div(A(x)Du)=H(x,Du) in Ω where A(x) is a bounded coercive matrix with measurable ingredients, λ≥0 and ξ↦H(x,ξ) has a super linear growth and is convex at infinity. We improve earlier results where the convexity of H(x,⋅) was required to hold globally.