Cauchy problem for higher order p-evolution equations

We consider p  -evolution equations of order m?2$m?2$ in (t,x)$(t,x)$ with real characteristics. We give sufficient conditions for the well-posedness of the Cauchy problem in Sobolev spaces, in terms of decay estimates of the coefficients as the space variable x→∞$x\to \infty$.     

Cauchy problem for p-evolution equations

We consider p-evolution equations with real characteristics. We give a condition, on the lower order terms, that is sufficient for well-posedness of the Cauchy problem in Sobolev spaces.     

A Liouville-type theorem for the p-Laplacian with potential term

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a singular p  -Laplacian problem with a potential term, such that a nonzero subsolution of another such problem is also a ground state. Unlike in the linear case (p=2$p=2$), this condition involves comparison of both the functions and of their gradients.     

Generalized Anti-Wick Operators with Symbols in Distributional Sobolev spaces

Generalized Anti-Wick operators are introduced as a class of pseudodifferential operators which depend on a symbol and two different window functions. Using symbols in Sobolev spaces with negative smoothness and windows in so-called modulation spaces, we derive new conditions for the boundedness on L² of such operators and for their membership in the Schatten classes. These results extend and refine results contained in [20], [10], [5], [4], and [14].     

Eigenvalue problems for the p-Laplacian

We study nonlinear eigenvalue problems for the p$p$-Laplace operator subject to different kinds of boundary conditions on a bounded domain. Using the Ljusternik–Schnirelman principle, we show the existence of a nondecreasing sequence of nonnegative eigenvalues. We prove the simplicity and isolation of the principal eigenvalue and give a characterization for the second eigenvalue.     

Almost minimizers for semilinear free boundary problems with variable coefficients

We study regularity results for almost minimizers of the functional where is a matrix with Hölder continuous coefficients. In the case we show that an almost minimizer belongs to , where the exponent η is related with the competition between the Hölder continuity of the matrix , the parameter of almost minimization and γ. In some sense, this regularity is optimal. As far as the case is concerned, our results show that an almost minimizer is locally in each phase and , improving in some sense a recent result of David & Toro.     

Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros

Using a combination of several methods, such as variational methods, the sub and supersolutions method, comparison principles and a priori estimates, we study existence, multiplicity, and the behavior with respect to λ of positive solutions of p-Laplace equations of the form −Δpu=λh(x,u), where the nonlinear term has p-superlinear growth at infinity, is nonnegative, and satisfies h(x,a(x))=0 for a suitable positive function a. In order to manage the asymptotic behavior of the solutions we extend a result due to Redheffer and we establish a new Liouville-type theorem for the p-Laplacian operator, where the nonlinearity involved is superlinear, nonnegative, and has positive zeros.     

Regularity for p-harmonic equations with right-hand side in Orlicz-Zygmund classes

We establish regularity results for solutions of some degenerate elliptic PDEs, with right-hand side in a suitable Orlicz-Zygmund class. The nonnegative function which measures the degree of degeneracy of the ellipticity bounds is assumed to be exponentially integrable. We find that the scale of improved regularity is logarithmic and we indicate its exact dependence on the degree of the degeneracy of the problem.     

Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains

We study the existence and nonexistence of positive (super)solutions to the nonlinear p-Laplace equation

     

Differentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg group

We propose a direct method to control the first-order fractional difference quotients of solutions to quasilinear subelliptic equations in the Heisenberg group. In this way we implement iteration methods on fractional difference quotients to obtain weak differentiability in the T-direction and then second-order weak differentiability in the horizontal directions.     

Zeros of Racah coefficients and the pell equation

The interface between Racah coefficients and mathematics is reviewed and several unsolved problems pointed out. The specific goal of this investigation is to determine zeros of these coefficients. The general polynomial is given whose set of zeros contains all nontrivial zeros of Racah (6j) coefficients [this polynomial is also given for the Wigner-Clebsch-Gordan (3j) coefficients]. Zeros of weight 1 3j- and 6j-coefficients are known to be related to the solutions of classic Diophantine equations. Here it is shown how solutions of the quadratic Diophantine equation known as Pell's equation are related to weight 2 zeros of 3j- and 6j-coefficients. This relation involves transformations of quadratic forms over the integers, the orbit classification of zeros of Pell's equation, and an algorithm for determining numerically the fundamental solutions of Pell's equation. The symbol manipulation program MACSYMA was used extensively to effect various factorings and transformations and to give a proof.The results of this paper were presented in an invited talk by one of us (JDL) at the NSF-CBMS Regional Conference on Special Functions, Physics and Computer Algebra, May 20–24, 1985, Arizona State University, Tempe, AZ.     

Smoothness of generalized solutions for higher-order elliptic equations with nonlocal boundary conditions

The smoothness of generalized solutions for higher-order elliptic equations with nonlocal boundary conditions is studied in plane domains. Necessary and sufficient conditions upon the right-hand side of the problem and nonlocal operators under which the generalized solutions possess an appropriate smoothness are established.     

LU- versus UL-Factorization of Integral Operators with Semi-Separable Kernel

The problem we consider is how to obtain a UL-factorization from a LU-factorization for integral operators with semi-separable kernels in both the time varying and the time invariant cases. We also consider the special situation where the integral operators are self-adjoint.     

The Navier-Stokes equations in the weak- LnL^n space with time-dependent external force

Abstract. We consider the Navier-Stokes equations with time-dependent external force, either in the whole time or in positive time with initial data, with domain either the whole space, the half space or an exterior domain of dimension . We give conditions on the external force sufficient for the unique existence of small solutions in the weak- space bounded for all time. In particular, this result gives sufficient conditions for the unique existence and the stability of small time-periodic solutions or almost periodic solutions. This result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak- space with time-independent external force. Received: 30 March 1999 / Accepted: 21 September 1999 / Published online: 28 June 2000     

Multiple positive solutions for nonlinear eigenvalue problems with the p-Laplacian

In this paper we consider a nonlinear eigenvalue problem driven by the p$p$-Laplacian differential operator and with a nonsmooth potential. Using degree theoretic arguments based on the degree map for certain operators of monotone type, we show that the problem has at least two nontrivial positive solutions as the parameter λ>0$\lambda >0$ varies in a half-line.     

Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols

We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an L^p,q space and M^∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.     

On sign-changing and multiple solutions of the p-Laplacian

In this paper, we construct the pseudo-gradient vector field in , by which we obtain the positive and negative cones of are both invariant sets of the descent flow of the corresponding functional. Then we use differential equations theory in Banach spaces and dynamics theory to study p-Laplacian boundary value problems with “jumping” nonlinearities at zero or infinity, and get new multiple solutions and sign-changing solutions theorems of p-Laplacian.     

On the fundamental eigenvalue ratio of the p-Laplacian

It is shown that the fundamental eigenvalue ratio of the p-Laplacian is bounded by a quantity depending only on the dimension N and p.     

The Muskat problem with surface tension and a nonregular initial interface

In this paper, we consider the two-dimensional Muskat problem with surface tension on a free boundary. The initial shape of the unknown interface has a corner point. We prove that the problem has a unique solution in the weighted Hölder classes locally in time and specify the sufficient conditions for the existence of the “waiting time” phenomenon.