1,721,038 research outputs found
Comparison principles for p-Laplace equations with lower order terms
No full textWe prove comparison principles for quasilinear elliptic equations whose simplest
model is
λu − Du + H(x, Du) = 0 ,
where Du = div (|Du|p−2Du) is the p-Laplace operator with p > 2, λ ≥ 0, H(x,ξ) :
× RN → R is a Carathéodory function and ⊂ 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
On the comparison principle for unbounded solutions of elliptic equations with first order terms
Full text linkWe 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
Large solutions and gradient bounds for quasilinear elliptic equations
No full textWe consider the quasilinear degenerate elliptic equation lambda u - Delta(p)u + H(x, Du) = 0 in Omega where (p) is the p-Laplace operator, p>2, 0 and is a smooth open bounded subset of (N) (N2). 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
Local and global time decay for parabolic equations with super linear first-order terms
No full textWe study a class of parabolic equations having first‐order terms with superlinear (and subquadratic) growth. The model problem is the so‐called viscous Hamilton–Jacobi equation with superlinear Hamiltonian. We address the problem of having unbounded initial data and we develop a local theory yielding well‐posedness for initial data in the optimal Lebesgue space, depending on the superlinear growth. Then we prove regularizing effects, short and long time decay estimates of the solutions. Compared to previous works, the main novelty is that our results apply to nonlinear operators with just measurable and bounded coefficients, since we totally avoid the use of gradient estimates of higher order. By contrast we only rely on elementary arguments using equi‐integrability, contraction principles and truncation methods for weak solutions
A Segregation Problem in Multi-Population Mean Field Games
No full textWe study a two-population mean field game in which the coupling
between the two populations becomes increasingly singular. In the case of a
quadratic Hamiltonian, we show that the limit system corresponds a partition of the
space into two components in which the players have to solve an optimal control
problem with state constraints and mean field interactions
The boundary behavior of blow-up solutions related to a stochastic control problem with state constraint
No full textWe consider solutions of the equation -Delta u + lambda u + |Delta u|(q) = f, which blow up uniformly at the boundary of a smooth domain, that can be interpreted as the value function of a state constraint control problem for a Brownian motion. We prove a complete asymptotic expansion of the gradient at the boundary, giving the precise behavior of normal and tangent components. The result is achieved by proving Lipschitz regularity for u - S, where S is an explicit singular corrector term. As the main motivation and application of our result, we characterize the behavior of the singular optimal control law and of the constrained dynamics near the boundary
Gradient bounds for elliptic problems singular at the boundary
No full textLet be a bounded smooth domain in \rn, ,
and let us denote by dist.
We study a class of singular Hamilton-Jacobi equations, arising from stochastic control problems, whose simplest model is
- \alpha \Delta u+ u + \frac{\D u \cdot B (x)}{d (x)}+ c(x) |\D u|^2=f (x) \quad \mbox{in } \Omega,
where belongs to and is (possibly) singular at , c\in \lip (with no sign condition) and the field B\in \lip^N has the outward direction and satisfies at ( 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.
In some cases, we show that this is the unique bounded solution.
We also discuss the stability of such estimates with respect to , as vanishes, obtaining Lipschitz solutions for first order problems with similar features.
The main tool is a refined weighted version of the classical
Bernstein's method to get gradient bounds; the key role is played here by the orthogonal transport component of the Hamiltonian
A variational approach to the mean field planning problem
Full text linkWe investigate a first-order mean field planning problem associated to a convex Hamiltonian H with quadratic growth and a monotone interaction term f with polynomial growth.
We exploit the variational structure of the system, which encodes the first order optimality condition of a convex dynamic optimal entropy-transport problem with respect to the unknown density m and of its dual, involving the maximization of an integral functional among all the subsolutions u of an Hamilton-Jacobi equation.
Combining ideas from optimal transport, convex analysis and renormalized solutions to the continuity equation, we will prove existence and (at least partial) uniqueness of a weak solution. A crucial step of our approach relies on a careful analysis of distributional subsolutions to Hamilton-Jacobi equations of the form , under minimal summability conditions on α, and to a measure-theoretic description of the optimality via a suitable contact-defect measure. Finally, using the superposition principle, we are able to describe the solution to the system by means of a measure on the path space encoding the local behavior of the players
Splitting methods and short time existence for the master equations in mean field games
Full text linkWe develop a splitting method to prove the well-posedness, in short time, of solutions for two master equations in mean field game (MFG) theory: the second order master equation, describing MFGs with a common noise, and the system of master equations associated with MFGs with a major player. Both problems are infinite-dimensional equations stated in the space of probability measures. Our new approach simplifies and generalizes previous existence results for second order master equations and provides the first existence result for systems associated with MFG problems with a major player
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