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31
ROBIN BOUNDARY VALUE PROBLEMS ON ARBITRARY DOMAINS
"... Abstract. We develop a theory of generalised solutions for elliptic boundary value problems subject to Robin boundary conditions on arbitrary domains, which resembles in many ways that of the Dirichlet problem. In particular, we establish LpLqestimates which turn out to be the best possible in tha ..."
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Cited by 27 (3 self)
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Abstract. We develop a theory of generalised solutions for elliptic boundary value problems subject to Robin boundary conditions on arbitrary domains, which resembles in many ways that of the Dirichlet problem. In particular, we establish LpLqestimates which turn out to be the best possible in that framework. We also discuss consequences to the spectrum of Robin boundary value problems. Finally, we apply the theory to parabolic equations. 1.
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 22 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Semigroups and Linear Partial Differential Equations with Delay
, 1999
"... We prove the equivalence of the wellposedness of a partial differential equation with delay and an associated abstract Cauchy problem. This is used to derive sufficient conditions for wellposedness, exponential stability and norm continuity of the solutions. Applications to a reactiondiffusion eq ..."
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Cited by 22 (0 self)
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We prove the equivalence of the wellposedness of a partial differential equation with delay and an associated abstract Cauchy problem. This is used to derive sufficient conditions for wellposedness, exponential stability and norm continuity of the solutions. Applications to a reactiondiffusion equation with delay are given.
L^pRegularity for Elliptic Operators with Unbounded Coefficients
, 2002
"... Under suitable conditions on the functions a 2 C N 2 ), F 2 C ), and V : R [0; 1), we show that the operator Au = r(aru) + F ru V u with domain W V (R ) = fu 2 ) : V u 2 L )g generates a positive analytic semigroup on L ), 1 < p < 1. ..."
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Cited by 12 (6 self)
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Under suitable conditions on the functions a 2 C N 2 ), F 2 C ), and V : R [0; 1), we show that the operator Au = r(aru) + F ru V u with domain W V (R ) = fu 2 ) : V u 2 L )g generates a positive analytic semigroup on L ), 1 < p < 1.
On the spectrum of Schrödinger operators with quasiperiodic algebrogeometric KdV potentials
 J. Analyse Math. 95
, 2005
"... Dedicated with great pleasure to Vladimir A. Marchenko on the occasion of his 80th birthday. Abstract. We characterize the spectrum of onedimensional Schrödinger operators H = −d 2 /dx 2 + V in L 2 (R; dx) with quasiperiodic complexvalued algebrogeometric potentials V (i.e., potentials V which s ..."
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Cited by 8 (4 self)
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Dedicated with great pleasure to Vladimir A. Marchenko on the occasion of his 80th birthday. Abstract. We characterize the spectrum of onedimensional Schrödinger operators H = −d 2 /dx 2 + V in L 2 (R; dx) with quasiperiodic complexvalued algebrogeometric potentials V (i.e., potentials V which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg–de Vries (KdV) hierarchy) associated with nonsingular hyperelliptic curves. The corresponding problem appears to have been open since the midseventies. The spectrum of H coincides with the conditional stability set of H and can explicitly be described in terms of the mean value of the inverse of the diagonal Green’s function of H. As a result, the spectrum of H consists of finitely many simple analytic arcs and one semiinfinite simple analytic arc in the complex plane. Crossings as well as confluences of spectral arcs are possible and discussed as well. These
Gaussian estimates and holomorphy of semigroups on L p spaces
 J. London Math. Soc
, 1996
"... Let O c U n be an open set and let Tp — (Tp(t))>0 be consistent semigroups on L P (Q) for 1 ^p < oo, with generators Ap. Assume that Tp is an analytic semigroup of angle <p for some />oe(l, oo). It is then natural to ask under which conditions the semigroups Tp are analytic too. For the ..."
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Cited by 6 (0 self)
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Let O c U n be an open set and let Tp — (Tp(t))>0 be consistent semigroups on L P (Q) for 1 ^p < oo, with generators Ap. Assume that Tp is an analytic semigroup of angle <p for some />oe(l, oo). It is then natural to ask under which conditions the semigroups Tp are analytic too. For the time being, suppose that Tp and Tp are
Heat kernel estimates and L^pspectral independence of elliptic operators
, 1997
"... Let\Omega be an open subset of IR d and let T p for p 2 [1; 1) be consistent C 0 semigroups given by kernels that satisfy an upper heat kernel estimate. Denoting by A p their generators, we show that the spectrum oe(A p ) is independent of p 2 [1; 1). We also treat the case of weighted L p spa ..."
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Cited by 6 (1 self)
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Let\Omega be an open subset of IR d and let T p for p 2 [1; 1) be consistent C 0 semigroups given by kernels that satisfy an upper heat kernel estimate. Denoting by A p their generators, we show that the spectrum oe(A p ) is independent of p 2 [1; 1). We also treat the case of weighted L p spaces for weights that satisfy a subexponential growth condition. An example shows that independence of the spectrum may fail for an exponential weight. We apply our result to Schrodinger operators, Petrovskij correct systems with Holder continuous coefficients, and elliptic operators in divergence form with real, but not necessarily symmetric coefficients and with complex coefficients. 1 Introduction and Main Results Let\Omega be an open subset of IR d and A be a closed linear operator in L 2(\Omega\Gamma := L 2(\Omega ; dx) where dx denotes Lebesgue measure. Assume that A generates a C 0 semigroup T in L 2(\Omega\Gamma which induces consistent C 0  semigroups T p with generators ...
Some classes of nonanalytic Markov semigroups
, 2004
"... We deal with Markov semigroups Tt corresponding to second order elliptic operators Au = ∆u +〈Du,F 〉, where F is an unbounded locally Lipschitz vector field on RN. We obtain new conditions on F under which Tt is not analytic in Cb(R N). In particular, we prove that the onedimensional operator Au = u ..."
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Cited by 4 (1 self)
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We deal with Markov semigroups Tt corresponding to second order elliptic operators Au = ∆u +〈Du,F 〉, where F is an unbounded locally Lipschitz vector field on RN. We obtain new conditions on F under which Tt is not analytic in Cb(R N). In particular, we prove that the onedimensional operator Au = u′′−x3u′, with domain {u ∈ C2(R): u, u′ ′ − x3u ′ ∈ Cb(R)}, is not sectorial in Cb(R). Under suitable hypotheses on the growth of F, we introduce a class of nonanalytic Markov semigroups in Lp(RN, µ), where µ is an invariant measure for Tt.