Results 1  10
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37
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 15 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
The Dirichlet problem in Lipschitz domains with boundary data in Besov spaces for higher order elliptic systems with rough coefficients
 J. Analyse Math
"... boundary data in Besov spaces for higher order elliptic systems with rough coefficients ∗ ..."
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Cited by 11 (5 self)
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boundary data in Besov spaces for higher order elliptic systems with rough coefficients ∗
Stationary boundary value problems for compressible Navier–Stokes equations
 Handbook of Differential Equations
, 2008
"... Abstract. In the paper compressible, stationary NavierStokes equations are considered. A framework for analysis of such equations is established. In particular, the wellposedness for inhomogeneous boundary value problems of elliptichyperbolic type is shown. Analysis is performed for small perturb ..."
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Cited by 8 (3 self)
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Abstract. In the paper compressible, stationary NavierStokes equations are considered. A framework for analysis of such equations is established. In particular, the wellposedness for inhomogeneous boundary value problems of elliptichyperbolic type is shown. Analysis is performed for small perturbations of the socalled approximate solutions, i.e., the solutions take form (1.12). The approximate solutions are determined from Stokes problem (1.11). The small perturbations are given by solutions to (1.13). The uniqueness of solutions for problem (1.13) is proved, and in addition, the differentiability of solutions with respect to the coefficients of differential operators is shown. The results on the wellposedness of nonlinear problem are interesting on its own, and are used to obtain the shape differentiability of the drag functional for incompressible NavierStokes equations. The shape gradient of the drag functional is derived in the classical and useful for computations form, an appropriate adjoint state is introduced to this end. The shape derivatives of solutions to the NavierStokes equations are given by smooth functions, however the shape differentiability is shown in a weak norm. The method of analysis proposed in the paper is general, and can be used to establish the wellposedness for distributed and boundary control problems as well as for inverse problems in the case of the state equations in the form of compressible NavierStokes equations. The differentiability of solutions to the NavierStokes equations with respect to the data leads to the first order necessary conditions for a broad class of optimization problems. 1.
Exact Smoothing Properties of Schrödinger Semigroups
, 1997
"... We study Schrodinger semigroups in the scale of Sobolev spaces, and show that, for Kato class potentials, the range of such semigroups in L p has exactly two more derivatives than the potential; this proves a conjecture of B. Simon. We show that eigenfunctions of Schrodinger operators are generica ..."
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Cited by 6 (0 self)
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We study Schrodinger semigroups in the scale of Sobolev spaces, and show that, for Kato class potentials, the range of such semigroups in L p has exactly two more derivatives than the potential; this proves a conjecture of B. Simon. We show that eigenfunctions of Schrodinger operators are generically smoother by exactly two derviatives (in given Sobolev spaces) than their potentials. We give applications to the relation between the potential's smoothness and particle kinetic energy in the context of quantum mechanics, and characterize kinetic energies in Coulomb systems. The techniques of proof invove Leibniz and chain rules for fractional derivatives which are of independent interest, as well as a new characterization of the Kato class. 1 Introduction In this paper we attempt a precise study of the action of Schrodinger semigroups in the scale of Sobolev spaces. Work in this area was begun by B. Simon [Si4] in 1985. He proved a number of positive and negative results on such smooth...
Multipliers between Sobolev spaces and fractionnal differentiation
 Jour. of Math. Analysis and Applications
"... differentiation ..."
Higher regularity in the classical layer potential theory for Lipschitz domains
 Indiana Univ. Math. J
"... Abstract. Classical boundary integral equations of the harmonic potential theory on Lipschitz surfaces are studied. We obtain higher fractional Sobolev regularity results for their solutions under weak conditions on the surface. These results are derived from a theorem on the solvability of auxilia ..."
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Cited by 5 (1 self)
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Abstract. Classical boundary integral equations of the harmonic potential theory on Lipschitz surfaces are studied. We obtain higher fractional Sobolev regularity results for their solutions under weak conditions on the surface. These results are derived from a theorem on the solvability of auxiliary boundary value problems for the Laplace equation in weighted Sobolev spaces. We show that classes of domains under consideration are optimal.