This paper is concerned with the state-constrained optimal control of the threedimensional thermistor problem, a fully quasilinear 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. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Prüss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. 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 in form of a qualified optimality system. The theoretical findings are illustrated by numerical results.for all ξ ∈ X. Moreover, in the proof of Theorem 4.5, ∂ θ B(θ, u) was found to be a topological isomorphism between the spaces X and L s (J; W −1,q ∅ ) × X s and consequently ∂ θ B(θ, u) * is also a topological isomorphism between L s (J; W 1,q ) × X s and X . In particular, for every f ∈ X there exists a unique p = p f ∈ L s (J; W 1,q ) × X s such that ∂ θ B(θ, u) * p = f. Hence, settinḡ
In this paper we establish Hölder estimates for solutions to nonautonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al. [40], which also serves as the starting point for our investigations. 2010 Mathematics Subject Classification. 35B65, 35K10, 35K15. Key words and phrases. Linear, second order parabolic equations, non-smooth data, Hölder continuity of the solution.147 148 HANNES MEINLSCHMIDT AND JOACHIM REHBERGdivergence of vector-valued L p -functions. As is well-known, such representations are highly non-unique; in particular the zero-functional may be represented as the divergence of a non-zero vector valued function. Lastly, it is not quite clear how broad the admissable geometric setting really is: on one hand "piecewise C 1 " is demanded, on the other the crucial "Condition A" ([40, pg. 9], compare also [39, Ch. II.B, Definition B.3]) -well-known from elliptic theory -comes into play.Our intention is to deliver a text which• clearly defines the underlying geometric concept for the domain Ω -thereby avoiding "Condition A", • incorporates mixed boundary conditions within an appropriately defined framework,• avoids a global Lipschitz condition for Ω -at least insofar the Dirichlet boundary part D is concerned, compare also [26], • gives a result in the formulation of classical Hölder spaces.The paper is organized as follows: In the next chapter we first introduce some terminology and our geometrical assumptions on the domain Ω and the Dirichlet boundary part D. Then, as a starting point, we quote the classical result on the existence and uniqueness of solutions for non-autonomous parabolic equations in a Hilbert spaces setting, namely the right-hand side f being taken from L 2 (J; V * ) yielding solutions from L 2 (J; V ) ∩ W 1,2 (J; V * ), where V → H → V * is a Gelfand triplet. Afterwards the main result is announced, cf. Theorem 2.13. In Ch. 3 we give the proof of this theorem, starting with the quotation of several classical results from the book [40]. The idea is to take these in a setting -namely the half cube and the ball and homogeneous Dirichlet boundary conditions -where the inherent technical difficulties are still not appearing: here it is clear that the general suppositions on the domain posed in [40] are fulfilled and, concerning the results, falls back to the classical Hölder spaces. After establishing some preliminaries in Ch. 3.2, in Ch. 3.3 we establish permanence principles, such as localization of the equation, bi-Lipschitz transformations and reflection which allow us to treat the problems in the geometric setting established above. This results in problems of the same complexity as before on a half-cube or a cube, respectively. Now employing the classical Ladyshenskaya results, one is enabled to deduce the required Hölder results for...
This work combines results from operator and interpolation theory to show that elliptic systems in divergence form admit maximal elliptic regularity on the Bessel potential scale H s−1 D (Ω) for s > 0 sufficiently small, if the coefficient in the main part satisfies a certain multiplier property on the spaces H s (Ω). Ellipticity is enforced by assuming a Gårding inequality, and the result is established for spaces incorporating mixed boundary conditions with very low regularity requirements for the underlying spatial set. To illustrate the applicability of our results, two examples are provided. Firstly, a phase-field damage model is given as a practical application where higher differentiability results are obtained as a consequence to our findings. These are necessary to show an improved numerical approximation rate. Secondly, it is shown how the maximal elliptic regularity result can be used in the context of quasilinear parabolic equations incorporating quadratic gradient terms.
Abstract. This paper is concerned with the state-constrained optimal control of the threedimensional thermistor problem, a fully quasilinear 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. Local existence, uniqueness and continuity for the state system as well as existence of optimal solutions, admitting global-in-time solutions, to the optimization problem were shown in the the companion paper of this work. In this part, we address further properties of the set of controls whose associated solutions exist globally such as openness, which includes analysis of the linearized state system via maximal parabolic regularity. 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 in form of a qualified optimality system in which we do not need to refer to the set of controls admitting global solutions. The theoretical findings are illustrated by numerical results. This work is the second of two papers on the three-dimensional thermistor problem.
In this paper we prove that the full Keller-Segel system, a quasilinear strongly coupled reaction-crossdiffusion system of four parabolic equations, is well-posed in space dimensions 2 and 3 in the sense that it always admits an unique local-in-time solution in an adequate function space, provided that the initial values are suitably regular. The proof is done via an abstract solution theorem for nonlocal quasilinear equations by Amann and is carried out for general source terms. It is fundamentally based on recent nontrivial elliptic and parabolic regularity results which hold true even on rather general nonsmooth spatial domains. This enables us to work in a nonsmooth setting which is not available in classical parabolic systems theory. Apparently, there exists no comparable existence result for the full Keller-Segel system up to now. Due to the large class of possibly nonsmooth domains admitted, we also obtain new results for the "standard" Keller-Segel system consisting of only two equations as a special case. AMS classification scheme numbers: 35A01, 35K45, 35K57, 35Q92, 92C17The full Keller-Segel model is well-posed on nonsmooth domains
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