Order Quasilinear Equations Research Articles

A Fefferman-Phong Type Inequality and Applications to Quasilinear Subelliptic Equations

We establish a nonlocal generalization of a well-known inequality by C. Fefferman and D. H. Phong $$\smallint _{\mathbb{R}^n } Vu^2 {\text{d}}x \leqslant C\smallint _{\mathbb{R}^n } \left| {Du} \right|^2 dx,$$ for u∈C 0 ∞ (ℝn) and V belonging to the Morrey space M with 1<s≤n/2, when the gradient in the right-hand side is replaced by the energy associated to an arbitrary system of Lipschitz continuous vector fields. Accordingly, the multiplier V is taken in an appropriate Morrey space defined using the Carnot–Caratheodory metric generated by the vector fields. As an application, we prove the Harnack inequality and the Holder continuity of solutions for a wide class of second order quasilinear subelliptic equations.

C∞ regularity of solutions of the Levi equation

We will prove the C∞ regularity of the classical solutions of the equation Lu=q(1+|gradu|2)3/21+ut2 where Lu=uxx+uyy+2uy-uxut1+ut2uxt-2ux+uyut1+ut2uyt+ux2+uy21+ut2utt,q∈C∞(Ω) and q(ξ) ≠ 0 for every ξ ∈ Ω. This is a second order quasilinear equation, whose characteristic form has zero determinant at every point, and for every function u. However we will write it as a sum of squares of nonlinear vector fields, and we will extablish the result by means of a suitable freezing method.

Positive solutions to a class of second order differential equations with singular nonlinearities

A class of second order quasilinear differential equations with singular nonlinearities is considered. The set of all possible solutions defined on a positive half-line [a,∞) is classified into six types according to their aymptotic behavior as t→∞, and sharp conditions are established for the existence of solutions belonging to each of the classified types.

Positive solutions of second order quasilinear equations corresponding to p-harmonic maps

Positive solutions of second order quasilinear equations corresponding to p-harmonic maps

On some equivalent properties of sub- and supersolutions in second order quasilinear elliptic equations

On some equivalent properties of sub- and supersolutions in second order quasilinear elliptic equations

Oscillation and Non-Oscillation Theorems for a Class of Second Order Quasilinear Difference Equations

In this paper there are established necesssary and sufficient conditions for the second order quasilinear difference equation \Delta(p_n \varphi (\Delta y_n)) + f(n,y_{n+1}) = 0 \ \ \ (n \in \mathbb N_0) to have various types of non-oscillatory solutions. In addition, in the case that the equation is either strongly superlinear or strongly sublinear, there are established necessary and sufficient conditions for all solutions to oscillate.

OSCILLATION THEORY FOR A CLASS OF SECOND ORDER QUASILINEAR DIFFERENCE EQUATIONS

In this paper the authors establish necessary and sufficient conditons for the second order quasilinear difference equation \[\Delta (\alpha_n(\Delta y_n)^{\alpha*})+f(n, y_{n+1})=0\qquad\qquad n\in N(n_0)\qquad\qquad \text{(E)}\] to have various types of nonoscillatory solutions. In addition, in the case when (E) is either strongly superlinear or strongly sublinear, they establish necessary and su伍 c1ent conditions for all solutions to oscillate.

Oscillation and nonoscillation criteria for second order quasilinear differential equations

Oscillation and nonoscillation criteria for second order quasilinear differential equations

Oscillation and nonoscillation theorems for a class of second order quasilinear functional-differential equations

Oscillation and nonoscillation theorems for a class of second order quasilinear functional-differential equations

FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES WITH SINGULAR INITIAL DATA Lp(p < ∞)

We study the existence problem for the equations of first order quasilinear equations in several independent variables with singular initial data Lp(p < ∞). We obtain the convergence of the Lp(p < ∞) bounded approximating sequences generated by the method of vanishing viscosity. The uniqueness of the generalized solutions which can be obtained by the method of vanishing viscosity is also obtained.

C ∞ regularity of solutions of the Levi equation inR 2n+1

We study the regularity of the solutions of the Levi equation in ℝ2n+1. It is a second order quasilinear equation whose characteristic matrix is positive semidefinite and has vanishing determinant at every point and for every functionu∈C2. We show that the operator associated to the equation can be represented as a sum of squares of non linear vector fields. Then, by using a freezing method, we prove theC∞ regularity of the solutions.

Nonoscillation theorems for second order quasilinear differential equations

Nonoscillation theorems for second order quasilinear differential equations

On strong precompactness of bounded sets of measure-valued solutions of a first order quasilinear equation

In this article it is proved that bounded sequences of measure-valued solutions of a non-degenerate first order quasilinear equation are precompact in the topology of strong convergence. The general case of flow functions which are merely continuous is considered.

Existence of positive entire solutions for higher order quasilinear elliptic equations

Existence of positive entire solutions for higher order quasilinear elliptic equations

Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy

Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy

Phragmen-Lindel�f type results for second order quasilinear parabolic equations in ?2

In this paper Phragmen-Lindelof type growth-decay estimates are derived for solutions of initial-boundary value problems associated with a class of quasilinear parabolic equations defined on a semi-infinite strip in ℝ2. The particular problems considered are ones in which homogeneous initial data and homogeneous Dirichlet conditions on the long sides of the strip are prescribed.

Global oscillatory waves for second order quasilinear wave equations

In this paper we prove the global existence and describe the asymptotic behaviour of a family of oscillatory solutions of Cauchy problems for a class of scalar second order quasilinear wave equations, when the space dimension is odd and at least equal to 3 3 . If time is bounded, corresponding results for quasilinear first order systems were obtained by Guès; to prove our results we reduce our problems to bounded time problems with the help of a conformal inversion. To obtain global results, suitable geometric assumptions must be made on the set where the oscillations are concentrated at initial time.

On similarity solutions of a third order quasilinear equation

Similarity solutions of a balance P.D.E. are described by a first order P.D.E., obtained by annihilating infinitesimal generator of isovector field of the closed ideal generated by the balance n- form of the equation and by the contact forms of the kinematic space. In this paper we apply similarity solutions theory to obtain a complete description of similarity solutions of a third order quasilinear equation which describes wave hierarchies in linear dissipative materials.

Blow–up of solutions for first order quasilinear hyperbolic equations

We investigate a class of scalar conservation laws with non uniformly convex flux function f = f (x, u). Under suitable assumptions on f , we prove the existence of weak entropy solutions which blow up in finite time. Uniqueness or multiplicity of such solutions is also studied.

Probabilistic interpretation of a system of quasilinear elliptic partial differential equations under Neumann boundary conditions

A probabilistic interpretation of a system of second order quasilinear elliptic partial differential equations under a Neumann boundary condition is obtained by introducing a kind of backward stochastic differential equations in the infinite horizon case. In the same time, a simple proof for the existence and uniqueness result of the classical solution of that Neumann problem is given.