C1,α Regularity of Viscosity Solutions of Fully Nonlinear Elliptic PDE Under Natural Structure Conditions

In this paper we are concemed with fully nonlinear elliptic equation F(x, u, Du, D²u) = 0. We establish the interior Lipschitz continuity and C^{1,α} regularity of viscosity solutions under natural structure conditions without differentiating the equation as usual, especially we give a new analytic Harnack inequality approach to C^{1,α} estimate for viscosity solutions instead of the geometric approach given by L. Caffarelli \& L. Wang and improve their results. Our structure conditions are rather mild.     

Ergodic Type Problems and Large Time Behaviour of Unbounded Solutions of Hamilton–Jacobi Equations

We study the large time behavior of Lipschitz continuous, possibly unbounded, viscosity solutions of Hamilton–Jacobi Equations in the whole space ? ^N . The associated ergodic problem has Lipschitz continuous solutions if the analogue of the ergodic constant is larger than a minimal value λ_min. We obtain various large-time convergence and Liouville type theorems, some of them being of completely new type. We also provide examples showing that, in this unbounded framework, the ergodic behavior may fail, and that the asymptotic behavior may also be unstable with respect to the initial data.     

Comparison Principle for Viscosity Solutions with High Growth at Infinity

This paper is concerned with the comparison principle for viscosity solutions of the nonlinear elliptic equation F(Du, D²u} + |u|^{s-1}u =f in R^N, where f is uniformly continuous and F satisfies some conditions about p (p > 2}. We got the comparison principle for the viscosity solutions with some high growth at infinity, which relies on the relation between p and s.     

Stability of solutions to parameterized nonlinear complementarity problems

n such that x≥0,  F(x,u)-v≥0 , and F(x,u)-v ^T·x=0 where these are vector inequalities. We characterize the local upper Lipschitz continuity of the (possibly set-valued) solution mapping which assigns solutions x to each parameter pair (v,u). We also characterize when this solution mapping is locally a single-valued Lipschitzian mapping (so solutions exist, are unique, and depend Lipschitz continuously on the parameters). These characterizations are automatically sufficient conditions for the more general (and usual) case where v=0. Finally, we study the differentiability properties of the solution mapping in both the single-valued and set-valued cases, in particular obtaining a new characterization of B-differentiability in the single-valued case, along with a formula for the B-derivative. Though these results cover a broad range of stability properties, they are all derived from similar fundamental principles of variational analysis. Received March 30, 1998 / Revised version received July 21, 1998 Published online January 20, 1999     

On solution sets of multivalued differential equations

Let X be a Banach space, 2^x\? the nonempty subsets of X,J = [o,a]?R and F:J×X→2^x\? a multivalued map. We consider U′ ? F(t,u) a.e. on J, u(o) = X_p ? X. A solution of (1) is understood to be a.e. differentiable with u′ Bochner integrable over J such that u(t) =X₀ + ∫^{0 t} u′(s)ds on J and u′(t)?F(t,u(t)) a.e. Under appropriate conditions on F the set S of solutions to (1) is compact ≠ ? in C_X (J), the space of continuous v : J → X with ∣v∣₀ = max∣v(t)∣. We concentrate on maps F with F(t,.) upper semicontinuous andshow that S is connected or even a compact R_δ in the sense of Borsuk. This is interesting in itself, but also in connection with the multivalued Poincare map in case F is periodic in time.     

Nonlinear scattering for some dispersive equations generalizing Benjamin-Bona-Mahony equation

The time decay of solutions to nonlinear dispersive equations of the typeMu _t+F(u)_x=0 is established using the optimal estimates for the linearized equation and standard techniques from scattering theory.     

A note on the Liouville method applied to elliptic eventually degenerate fully nonlinear equations governed by the Pucci operators and the Keller–Osserman condition

This paper is devoted to the study of the viscosity solutions of

The Uniqueness of Viscosity Solutions of the Second Order Fully Nonlinear Elliptic Equations

Recently R. Jensen [1] has proved the uniqueness of viscosity solutions in W^{1,∞} of second order fully nonlinear elliptic equation F (D², Du, u) = 0. He does not assume F to be convex. In this paper we extend his result [1] to the case that F can be dependent on x, i. e. prove that the viscosity solutions in W^{1,∞} of the second order fully nonlinear elliptic equation F (D²u, Du, u, x) = 0 are unlique. We do not assume F to be convex either.     

Semicontinuous solutions for Hamilton-Jacobi equations and theL ∞control problem

We prove uniqueness for extended real-valued lower semicontinuous viscosity solutions of the Bellman equation forL -control problems. This result is then used to prove uniqueness for lsc solutions of Hamilton-Jacobi equations of the form –u _t +H(t, x, u, –Du)=0, whereH(t, x, r, p) is convex inp. The remaining assumptions onH in the variablesr andp extend the currently known results.Supported in part by Grant DMS-9300805 from the National Science Foundation.     

A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains

Let u be a vector field on a bounded Lipschitz domain in ?³, and let u together with its divergence and curl be square integrable. If either the normal or the tangential component of u is square integrable over the boundary, then u belongs to the Sobolev space H^1/2 on the domain. This result gives a simple explanation for known results on the compact embedding of the space of solutions of Maxwell's equations on Lipschitz domains into L².     

An integrable equation with nonsmooth solitons

We present the bi-Hamiltonian structure and Lax pair of the equation ρ_t = bu_x+(1/2)[(u ² −u_x ² )ρ]_x, where ρ = u − u_xx and b = const, which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show that under the vanishing boundary condition u → 0 at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions.     

Phénomène d’explosion totale pour l’équation de la chaleur avec non-linéarité du type puissance

In this Note, we are interested in the possible continuation after the blow-up time T_m of radially symmetric positive classical solutions u of the heat equation with nonlinearity f(u) = u^p, where p > 1. We say that u blows up completely after T_m if u can not be extended beyond T_m (even in the weak sense). We obtain a complete blow up criterion which relies on the asymptotic behaviour of u around the blow-up singularity x = 0.     

Bounded mild solutions to fractional integro-differential equations in Banach spaces

We study the existence and uniqueness of bounded solutions for the semilinear fractional differential equation $$D^\alpha u(t)= Au(t)+ \int_{-\infty}^t a(t-s)Au(s)ds+ f \bigl(t,u(t) \bigr), \quad t \in\mathbb{R}, $$ where A is a closed linear operator defined on a Banach space X, α>0, a∈L ¹(?₊) is a scalar-valued kernel and f:?×X→X satisfies some Lipschitz type conditions. Sufficient conditions are established for the existence and uniqueness of an almost periodic, almost automorphic and asymptotically almost periodic solution, among other.     

On a class of nonlinear fourth order differential equations

Summary In this paper we study the nonlinear fourth order differentiai equation u^iv±F(u, v)u=0, where F is a pssitive monotone function of u. Asymptotic behaviour of certain solutions are treated in sections 2 and 4 while a two point boundary value problem is studied in section 3. Entrata in Redazione il 26 agosto 1968.     

Convergence of approximate solutions to an elliptic–parabolic equation without the structure condition

We study the Cauchy–Dirichlet problem for the elliptic–parabolic equation $$b(u)_t + {\rm div} F(u) - \Delta u = f$$ in a bounded domain. We do not assume the structure condition $$b(z) = b(\hat z) \Rightarrow F(z) = F(\hat z).$$ Our main goal is to investigate the problem of continuous dependence of the solutions on the data of the problem and the question of convergence of discretization methods. As in the work of Ammar and Wittbold (Proc R Soc Edinb 133A(3):477–496, 2003) where existence was established, monotonicity and penalization are the main tools of our study. In the case of a Lipschitz continuous flux F, we justify the uniqueness of u (the uniqueness of b(u) is well-known) and prove the continuous dependence in L ¹ for the case of strongly convergent finite energy data. We also prove convergence of the ${\varepsilon}$ -discretized solutions used in the semigroup approach to the problem; and we prove convergence of a monotone time-implicit finite volume scheme. In the case of a merely continuous flux F, we show that the problem admits a maximal and a minimal solution.     

Solutions of multidimensional partial differential equations representable as a one-dimensional flow

We propose an algorithm for reducing an (M+1)-dimensional nonlinear partial differential equation (PDE) representable in the form of a one-dimensional flow u_t + $w_{x_1 } $ (u, u_x u_xx,…) = 0 (where w is an arbitrary local function of u and its xi derivatives, i = 1,…, M) to a family of M-dimensional nonlinear PDEs F(u,w) = 0, where F is a general (or particular) solution of a certain second-order two-dimensional nonlinear PDE. In particular, the M-dimensional PDE might turn out to be an ordinary differential equation, which can be integrated in some cases to obtain explicit solutions of the original (M+1)-dimensional equation. Moreover, a spectral parameter can be introduced in the function F, which leads to a linear spectral equation associated with the original equation. We present simplest examples of nonlinear PDEs together with their explicit solutions.     

Symmetries and Critical Phenomena in Fluids

We study two-dimensional active scalar systems arising in fluid dynamics in critical spaces in the whole plane. We prove an optimal well-posedness result that allows for the data and solutions to be scale-invariant. These scale-invariant solutions are new and their study seems to have far-reaching consequences. More specifically, we first show that the class of bounded vorticities satisfying a discrete rotational symmetry is a global existence and uniqueness class for the two-dimensional Euler squation. That is, in the well-known L¹∩L^∞ theory of Yudovich, the L¹-assumption can be dropped upon having an appropriate symmetry condition. We also show via explicit examples the necessity of discrete symmetry for the uniqueness. This already answers problems raised by Lions in 1996 and Bendetto, Marchioro, and Pulvirenti in 1993. Next, we note that merely bounded vorticity allows for one to look at solutions that are invariant under scaling—the class of vorticities that are 0-homo-geneous in space. Such vorticity is shown to satisfy a new one-dimensional evolution equation on 𝕊¹. Solutions are also shown to exhibit a number of interesting properties. In particular, using this framework, we construct time quasi-periodic solutions to the two-dimensional Euler equation exhibiting pendulum-like behavior. Finally, using the analysis of the one-dimensional equation, we exhibit strong solutions to the two-dimensional Euler equation with compact support for which angular derivatives grow at least (almost) quadratically in time (in particular, superlinear) or exponential in time (the latter being in the presence of a boundary). A similar study can be done for the surface quasi-geostrophic (SQG) equation. Using the same symmetry condition, we prove local existence and uniqueness of solutions that are merely Lipschitz continuous near the origin—though, without the symmetry, Lipschitz initial data is expected to lose its Lipschitz continuity immediately. Once more, a special class of radially homogeneous solutions is considered, and we extract a one-dimensional model that bears great resemblance to the so-called De Gregorio model. We then show that finite-time singularity formation for the one-dimensional model implies finite-time singularity formation in the class of Lipschitz solutions to the SQG equation that are compactly support. While the study of special infinite energy (i.e., nondecaying) solutions to fluid models is classical, this appears to be the first case where these special solutions can be embedded into a natural existence/uniqueness class for the equation. Moreover, these special solutions approximate finite-energy solutions for long time and have direct bearing on the global regularity problem for finite-energy solutions. © 2019 Wiley Periodicals, Inc.     

A theorem of global existence of solutions to nonlinear wave equations in four space dimensions

In this paper, the following nonlinear wave equation is considered; □u=F(u, D u, D _x D u),x∈R ⁿ ,t>0; u=u ⁰ (x), u _t =u ¹ (x), t=0. We prove that if the space dimensionn ≥ 4 and the nonlinearityF is smooth and satisfies a mild condition in a small neighborhood of the origin, then the above problem admits a unique and smooth global solution (in time) whenever the initial data are small and smooth. The strategy in proof is to use and improve Global Sobolev Inequalities in Minkowski space (see [8]), and to develop a generalized energy estimate for solutions.     

Graphs with prescribed Levi form trace

Abstract We prove existence and uniqueness of a viscosity solution of the Dirichlet problem related to the prescribed Levi mean curvature equation, under suitable assumptions on the boundary data and on the Levi curvature of the domain. We also show that such a solution is Lipschitz continuous by proving that it is the uniform limit of a sequence of classical solutions of elliptic problems and by building Lipschitz continuous barriers. Keywords: Levi mean curvature, Quasilinear degenerate elliptic PDE’s, Viscosity solutions, Comparison principle, Global Lipschitz estimates