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Isabeau Birindelli Giulio Galise Hitoshi Ishii 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2018,35(2):417-441
In this paper we investigate the validity and the consequences of the maximum principle for degenerate elliptic operators whose higher order term is the sum of k eigenvalues of the Hessian. In particular we shed some light on some very unusual phenomena due to the degeneracy of the operator. We prove moreover Lipschitz regularity results and boundary estimates under convexity assumptions on the domain. As a consequence we obtain the existence of solutions of the Dirichlet problem and of principal eigenfunctions. 相似文献
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We develop a theory of complex powers of the generalized Klein-Gordon-Fock operator . The negative powers of this operator are realized as potential-type integrals with nonstandard metrics, while positive powers inverse to negative ones are realized as approximative inverse operators.
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$m^2 - \square - i\lambda \frac{{\partial ^2 }}{{\partial x_1^2 }},\lambda > 0.$
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In the L p -spaces, we study the complex powers of the operator where δ is the Laplace operator. The complex powers G λ ?α/2 , Reα > 0, are realized as potential type operators B λ α with a nonstandard metric. We obtain L p → L p + L s -estimates for the operator B λ α . By using the method of approximate inverse operators, we construct the inversion of the potentials B λ α φ with L p -densities and describe the range B λ α (L p ) in terms of the inversion constructions.
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$G_\lambda = m^2 I + \Delta - i\lambda \frac{{\partial ^2 }}{{\partial x_1^2 }},0 < \lambda < 1,m > 0,$
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In this paper, we establish a Lebeau–Robbiano spectral inequality for a degenerate one-dimensional elliptic operator, and we show how it can be used to study impulse control and finite-time stabilization for a degenerate parabolic equation. 相似文献
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Kevin R. Payne 《Annali di Matematica Pura ed Applicata》2006,185(4):613-625
For partial differential equations of mixed elliptic-hyperbolic and degenerate types which are the Euler-Lagrange equations
for an associated Lagrangian, we examine an associated metric structure which becomes singular on the hypersurface where the
operator degenerates. In particular, we show that the “non-trivial part” of the complete symmetry group for the differential
operator (calculated in a previous paper by Lupo, D., Payne, K.R. [Conservation laws for equations of mixed elliptic-hyperbolic
and degenerate types. Duke Math. J. [15]]) corresponds to a group of local conformal transformations with respect to the metric
away from the metric singularity and that the group extends smoothly across the singular surface. In this way, we define and
calculate the conformal group for these operators as well as for lower order singular perturbations which are defined naturally
by the singular metric.
Mathematics Subject Classification (2000) 35M10, 58J70, 53A30
Work supported by MIUR, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari” and MIUR, Project “Metodi Variazionali
e Topologici nello Studio di Fenomeni Non Lineari”. 相似文献
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Second-order operators with degenerate coefficients 总被引:1,自引:0,他引:1
ter Elst A. F. M.; Robinson Derek W.; Sikora Adam; Zhu Yueping 《Proceedings London Mathematical Society》2007,95(2):299-328
We consider properties of second-order operators on d with bounded real symmetric measurable coefficients.We assume that C = (cij) 0 almost everywhere, but allow forthe possibility that C is degenerate. We associate with H acanonical self-adjoint viscosity operator H0 and examine propertiesof the viscosity semigroup S(0) generated by H0. The semigroupextends to a positive contraction semigroup on the Lp-spaceswith p [1, ]. We establish that it conserves probability andsatisfies L2 off-diagonal bounds, and that the wave equationassociated with H0 has finite speed of propagation. Nevertheless,S(0) is not always strictly positive because separation of thesystem can occur even for subelliptic operators. This demonstratesthat subelliptic semigroups are not ergodic in general and theirkernels are neither strictly positive nor Hölder continuous.In particular, one can construct examples for which both upperand lower Gaussian bounds fail even with coefficients in C2–(d)with > 0. 相似文献
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V. V. Pospelov 《Mathematical Notes》1977,22(2):610-613
The definition of the degree of a family of extension operators is given. The connection of lower estimates with general properties is established. The problem of approximation of the segment [0, 1] by finite subsets is studied. In the class of homogeneous operators the independence of norms of operators from the degree of the family of extension operators is proved.Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 215–219, August, 1977. 相似文献
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M. Dalla Riva J. Morais P. Musolino 《Mathematical Methods in the Applied Sciences》2013,36(12):1569-1582
The purpose of this paper is to construct a family of fundamental solutions for elliptic partial differential operators with quaternion constant coefficients. The elements of such family are expressed by means of functions, which depend jointly real analytically on the coefficients of the operators and on the spatial variable. We show some regularity properties in the frame of Schauder spaces for the corresponding single layer potentials. Ultimately, we exploit our construction by showing a real analyticity result for perturbations of the layer potentials corresponding to complex elliptic partial differential operators of order two. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
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C Hoel 《Journal of Differential Equations》1974,15(3):379-417
In this paper we modify the usual series computation for a parametrix of an elliptic operator and find a method that is applicable to some simple degenerate operators. In fact, we are able to construct fundamental solutions for operators such as Dx + ixDy and Dx2 + x2Dy2 + cDy which explain the lack of local solvability in a natural way. 相似文献
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Let ?? be an open subset of R d and ${ K=-\sum^d_{i,j=1}\partial_i\,c_{ij}\,\partial_j+\sum^d_{i=1}c_i\partial_i+c_0}$ a second-order partial differential operator with real-valued coefficients ${c_{ij}=c_{ji}\in W^{1,\infty}_{\rm loc}(\Omega),c_i,c_0\in L_{\infty,{\rm loc}}(\Omega)}$ satisfying the strict ellipticity condition ${C=(c_{ij}) >0 }$ . Further let ${H=-\sum^d_{i,j=1} \partial_i\,c_{ij}\,\partial_j}$ denote the principal part of K. Assuming an accretivity condition ${C\geq \kappa (c\otimes c^{\,T})}$ with ${\kappa >0 }$ , an invariance condition ${(1\!\!1_\Omega, K\varphi)=0}$ and a growth condition which allows ${\|C(x)\|\sim |x|^2\log |x|}$ as |x| ?? ?? we prove that K is L 1-unique if and only if H is L 1-unique or Markov unique. 相似文献
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P. T. Thuy N. M. Tri 《NoDEA : Nonlinear Differential Equations and Applications》2013,20(3):1213-1224
In this paper we consider initial boundary value problem for semilinear parabolic equations involving strongly degenerate elliptic differential operators. Depending on the concrete types of nonlinearity we establish the existence of compact connected global attractors of semigroups generated by the problem under consideration. 相似文献
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This paper is devoted to the study of the following degenerate Neumann problem for a quasilinear elliptic integro-differential operator Here is a second-order elliptic integro-differential operator of Waldenfels type and is a first-order Ventcel' operator with a(x) and b(x) being non-negative smooth functions on such that on . Classical existence and uniqueness results in the framework of H?lder spaces are derived under suitable regularity and structure conditions on the nonlinear term f(x,u,Du). Received April 22, 1997; in final form March 16, 1998 相似文献
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Roberto Stasi 《NoDEA : Nonlinear Differential Equations and Applications》2006,12(4):419-436
In this paper we are interested in studying the properties of an elliptic degenerate operator N0 in the space Lp of
with respect to an invariant measure μ. The existence of μ is proven under suitable conditions on coefficients of the operator.
We prove that the closure of N0 is m-dissipative in
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