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- Partial differential equations (4)
- dissipative Schroedinger-type operators (4)
- discontinuous coefficients (3)
- nonsmooth domains (3)
- Elliptic transmission problems (2)
- Lax-Phillips scattering theory (2)
- Sturm-Liouville operators (2)
- carrier and current densities (2)
- characteristic function (2)
- density matrices (2)
We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.
The paper is devoted to Schroedinger operators on bounded intervals of the real axis with dissipative boundary conditions. In the framework of the Lax-Phillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed, in particular, trace formula and Birman-Krein formula are verified directly. The results are used for dissipative Schroedinger-Poisson systems.
The well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is re-established for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented.
We regard drift-diffusion equations for semiconductor devices in Lebesgue spaces. To that end we reformulate the (generalized) van Roosbroeck system as an evolution equation for the potentials to the driving forces of the currents of electrons and holes. This evolution equation falls into a class of quasi-linear parabolic systems which allow unique, local in time solution in certain Lebesgue spaces. In particular, it turns out that the divergence of the electron and hole current is an integrable function. Hence, Gauss' theorem applies, and gives the foundation for space discretization of the equations by means of finite volume schemes. Moreover, the strong differentiability of the electron and hole density in time is constitutive for the implicit time discretization scheme. Finite volume discretization of space, and implicit time discretization are accepted custom in engineering and scientific computing. ---This investigation puts special emphasis on non-smooth spatial domains, mixed boundary conditions, and heterogeneous material compositions, as required in electronic device simulation.
Let H be a semi–bounded self–adjoint operator in a separable Hilbert space.
For a certain class of positive, continuous, decreasing, and convex functions
F we show the convexity of trace functionals tr(F (H + U − ε(U ))) − ε(U ),
where U is a bounded self–adjoint operator on H and ε(U ) is a normalizing
real function—the Fermi level—which may be identical zero. If additionally
F is continuously differentiable, then the corresponding trace functional is
Fréchet differentiable and there is an expression of its gradient in terms off
the derivative of F . The proof of the differentiability of the trace functional
is based upon Birman and Solomyak’s theory of double Stieltjes operator
integrals. If, in particular, H is a Schrödinger–type operator and U a real-valued function, then the gradient of the trace functional is the quantum
mechanical expression of the particle density with respect to an equilibrium
distribution function f = −F . Thus, the monotonicity of the particle density
in its dependence on the potential U of Schrödinger’s operator—which has
been understood since the late 1980s—follows as a special case.
We study in detail Schroedinger-type operators on a bounded interval of the real axis with dissipative boundary conditions. The characteristic function of such operators is computed, its minimal self-adjoint dilation is constructed and the generalized eigenfunction expansion for the dilation is developed. The problem is motivated by semiconductor physics.
We describe an embedding of a quantum mechanically described structure into a macroscopic flow. The open quantum system is partly driven by an adjacent macroscopic flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. This leads to an essentially non-selfadjoint Schroedinger-type operator, the spectral properties of which will be investigated.
Using a classical theorem of Sobolevskii on equations of parabolic type in a Banach space and recently obtained results on elliptic operators with discontinuous coefficients including mixed boundary conditions we prove that quasilinear parabolic systems in diagonal form admit a local, classical solution in the space of p-integrable functions, for some p greater than 1, over a bounded two dimensional space domain. As applications we have in mind systems of reaction diffusion equations, e.g. van Roosbroeck's system. The treatment of such equations in a space of integrable functions enables us to define the normal component of the flow across any part of the Dirichlet boundary by Gauss' theorem.
This paper is concerned with the state-constrained optimal control of the
two-dimensional thermistor problem, a quasi-linear coupled system
of a parabolic and elliptic PDE with mixed boundary conditions.
This system models the heating of a conducting material by means of direct current.
Existence, uniqueness and continuity for the state system are derived by employing
maximal elliptic and parabolic regularity. By similar arguments the
linearized state system is discussed, while the adjoint system involving measures
is investigated using a duality argument. These results allow to derive
first-order necessary conditions for the optimal control problem.
We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the Robin boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an Lp function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.
For rather general thermodynamic equilibrium distribution functions the density of a statistical ensemble of quantum mechanical particles depends analytically on the potential in the Schrödinger operator describing the quantum system. A key to the proof is that the resolvent to a power less than one of an elliptic operator with non-smooth coefficients, and mixed Dirichlet/Neumann boundary conditions on a bounded up to three-dimensional Lipschitz domain factorizes over the space of essentially bounded functions.
We prove an optimal regularity result for elliptic operators $-\nabla \cdot \mu \nabla:W^1,q_0 \rightarrow W^-1,q$ for a $q>3$ in the case when the coefficient function $\mu$ has a jump across a $C^1$ interface and is continuous elsewhere. A counterexample shows that the $C^1$ condition cannot be relaxed in general. Finally, we draw some conclusions for corresponding parabolic operators.
Let $\Upsilon$ be a three-dimensional Lipschitz polyhedron, and assume that the matrix function $\mu$ is piecewise constant on a polyhedral partition of $\Upsilon$. Based on regularity results for solutions to two-dimensional anisotropic transmission problems near corner points we obtain conditions on $\mu$ and the intersection angles between interfaces and $\partial \Upsilon$ ensuring that the operator $-\nabla \cdot \mu \nabla$ maps the Sobolev space $W^1,q_0(\Upsilon)$ isomorphically onto $W^-1,q(\Upsilon)$ for some $q > 3$.
Optimality Conditions for State-Constrained PDE Control Problems with Time-Dependent Controls
(2008)
The paper deals with optimal control problems for semilinear
elliptic and parabolic PDEs subject to pointwise state constraints.
The main issue is that the controls are taken from a restricted
control space. In the parabolic case, they are vector-valued
functions of the time, while they are vectors in elliptic
problems. Under natural assumptions, first- and second-order
sufficient optimality conditions are derived. The main result is the
extension of second-order sufficient conditions to semilinear
parabolic equations in domains of arbitrary dimension. In the
elliptic case, the problems can be handled by known results of
semi-infinite optimization. Here, different examples are discussed
that exhibit different forms of active sets and where second-order
sufficient conditions are satisfied at the optimal solution.
An one-dimensional Kohn-Sham system for spin particles is considered which effectively describes semiconductor nanostructures and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schrödinger operator with effective Kohn-Sham potential and obtain $W^1,2$-bounds of the associated particle density operator. Afterwards, compactness and continuity results allow to apply Schauder's fixed point theorem. In case of vanishing exchange-correlation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero.
For an operator-valued block-matrix model, which is called in quantum physics a Feshbach decomposition, a scattering theory is considered. Under trace class perturbation the channel scattering matrices are calculated. Using Feshbach's optical potential it is shown that for a given spectral parameter the channel scattering matrices can be recovered either from a dissipative or from a Lax-Phillips scattering theory.
A 1D coupled drift-diffusion dissipative Schroedinger model (hybrid model), which
is capable to describe the transport of electrons and holes in semi-conductor devices
in a non-equilibrium situation, is mathematically analyzed. The device domain is
split into a part where the transport is well-described by the drift-diffusion equations
(classical zone) and a part where a quantum description via a dissipative Schroedinger
system (quantum zone) is used. Both system are coupled such that the continuity
of the current densities is guaranteed. The electrostatic potential is self-consistently
determined by Poisson's equation on the whole device. We show that the hybrid
model is well-posed, prove existence of solutions and show their uniform boundedness
provided the distribution function satisfy a so-called balance condition. The current
densities are different from zero in the non-equilibrium case and uniformly bounded.
We study a stationary Schrödinger-Poisson system on a bounded interval of the real axis. The Schrödinger operator is defined on the bounded domain with transparent boundary conditions. This allows us to model a non-zero current through the boundary of the interval. We prove that the system always admits a solution and give explicit a priori estimates for the solutions.