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Quasilinear nonsmooth evolution systems in L^p-spaces on three-dimensional domains

Collaborator: J. Elschner (FG 4), J. Rehberg (FG 1), G. Schmidt

(FG 4)

Cooperation with: V. Maz'ya (Linköping University, Sweden)

Description: The reported work continues efforts to prove existence, uniqueness, and regularity results for elliptic and parabolic equations and systems which describe phenomena in physics, chemistry, and biology (see Annual Research Reports 2000 p. 19 et sqq. and 2002 p. 39 et sqq.). In particular, we are interested in quasilinear systems of the form

$$\nabla \mu_{k}^{} \nabla \nabla$$ (1)

which comprise--among others--reaction-diffusion systems and heat conduction, see [1] or [4] and the references cited therein. The focus is on the case of spatially three-dimensional nonsmooth domains and discontinuous coefficients $\mu_{k}^{}$, which occur in modeling heterogeneous media. Over the past years various tools for the study of such equations have been developed at the Weierstrass Institute, see [13, 14, 15, 19]. In particular, the result of Gröger [15] is of great use in many applications, see [2, 4, 7, 11, 18] to name only a few. Gröger's result states that

$$\nabla \mu \nabla \Gamma \Omega \mapsto \Gamma \Omega$$ (2)

is a topological isomorphism for q $\in$ ]2, q₀[ in case of Lipschitz domains, (elliptic) L^$\infty$ coefficient functions, and mixed boundary conditions. Unfortunately, it is well known that in general q₀ exceeds 2 only slightly. Thus, in view of embedding theorems generically only the case of two space dimensions is covered. However, for many applications it is not sufficient to study only equations in two space dimensions, that means physical systems which are--in one space direction--translational or circular invariant, see, e.g., [2] or [9]. The increasing structural complexity of technical devices requires to perform simulations and the corresponding mathematical analysis on three-dimensional domains, see [10, 16].

In our recent paper [22] we study the Dirichlet problem for (1) with piecewise constant coefficients $\mu_{k}^{}$ in a polyhedral domain $\Omega$ $\subset$ $\IR^{3}_{}$. We give conditions under which the problem admits a unique solution from a space

C([T₀, T], L^p($$\Omega$$;$$\IR^{m}_{}$$)) $$\cap$$ C¹((T₀, T], L^p($$\Omega$$;$$\IR^{m}_{}$$)).

The Dirichlet boundary data may depend on time, and $\Omega$ is a Lipschitz polyhedron, that means $\Omega$ is a bounded Lipschitz domain with piecewise plane boundary. Furthermore, we assume that $\Omega$ is the union of a finite number of Lipschitz polyhedra $\Omega_{1}^{}$, ..., $\Omega_{l}^{}$ such that the (3 x 3) matrix functions $\mu_{k}^{}$ are constant on these subdomains. The dependence of the functions R_k on $\nabla$u is not stronger than quadratic. The main advantage of our approach in comparison to the concept of weak solutions is the strong differentiability of the solution with respect to time and that the divergence of the corresponding currents j_k = $\mu_{k}^{}$J_k(t,u)$\nabla$u_k are functions, not only distributions. In a strict sense, only this justifies the application of Gauss' theorem to calculate the normal components of the currents over boundaries of (suitable) subdomains. Our main result, [22, Theorem 6.10], ensures the continuity of the normal fluxes across interfaces. This property is also very important in the numerical analysis of finite volume methods for heterostructures.

The local existence result for (1) rests upon the classical theorem of Sobolevskii on abstract quasilinear parabolic equations in Banach spaces and estimates for elliptic transmission problems. The problem is to find an adequate function space with respect to which the hypotheses of Sobolevskii's theorem can be verified. In the three-dimensional case one has to ensure that the linear operators in (2) are topological isomorphisms for some q > 3, if the matrices $\mu$ = $\mu_{k}^{}$ in (1) are piecewise constant. The operator (2) corresponds to an interface (or transmission) problem for the Laplacian, with different anisotropic materials given on the polyhedral subdomains $\Omega_{1}^{}$, ..., $\Omega_{l}^{}$ of $\Omega$, with Dirichlet conditions given on $\partial$$\Omega$.

In contrast to the pure Laplacian on a Lipschitz domain, see [17, Theorem 0.5], the gradients of solutions to the transmission problems only belong to L^{2+$\varepsilon$} for some $\varepsilon$ > 0. In the vicinity of vertices and edges, $\varepsilon$ may be arbitrarily small, even for polygonal interface problems with only four isotropic materials meeting in a vertex, see [20]. In the case of complex material coefficients, which corresponds to some special anisotropy, even two materials can produce strong singularities near vertices (see [5, 6], where similar problems are studied for Helmholtz equations). Therefore, a large part of our investigation, [22], is devoted to the optimal regularity for (2). This result inherently applies to elliptic systems describing heterostructures on three-dimensional domains.

It is well known that the singularities of solutions to elliptic boundary value problems near vertices and edges can be characterized in terms of the eigenvalues of certain polynomial operator pencils on domains of the unit sphere or the unit circle. We refer to [21] for the case of the Dirichlet and Neumann problem and to [12] for the polyhedral Laplace interface problem with two isotropic materials. The corresponding analysis for several anisotropic materials has not been performed so far and is the subject of our investigation in [22].

To avoid the cumbersome analysis of optimal regularity near vertices, see [3], we use the somewhat surprising fact that if the solution of the interface problem belongs to L^q for some q > 3 near each interior point of the interface and boundary edges, then the operator (2) is an isomorphism. Thus, the regularity result for (2) can be reduced to that for an interface problem on dihedral angles with one common edge. The proof relies essentially on sharp pointwise estimates of Green's function, which we perform in [22].

The main result of our linear regularity theory is that the operator (2) is an isomorphism for some q > 3 provided that a parameter $\widehat{\lambda}_{{\Omega}}^{}$, which depends on the decomposition of $\Omega$ into the subdomains $\Omega_{j}^{}$, satisfies the inequality

$$\widehat{\lambda}_{{\Omega}}^{} {\frac{{1}}{{3}}}$$ (3)

This number is the minimum over all edges of spectral parameters, which can be expressed in terms of the eigenvalues of certain transmission problems on the unit circle. These problems are obtained applying the partial Fourier transform along an edge and the Mellin transform with respect to the radial direction. The regularity result is sufficient for the treatment of the quadratic gradient terms in (1), if the Banach space is the space L^p with p = q/2. However, condition (3) imposes a rather strong assumption on the geometry of the subdomains $\Omega_{j}^{}$ and the coefficient $\mu_{k}^{}$, or equivalently, on the eigenvalues of certain pencils of ordinary differential operators. These conditions can be checked for many heterostructures of practical interest. Though at this point our results are restricted to Dirichlet boundary conditions, it should be possible to extend the result to mixed boundary conditions, which occur, e.g., in modeling semiconductor devices ([8]). This problem will be investigated in 2004.

References:

1. H. AMANN, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in: Function Spaces, Differential Operators and Nonlinear Analysis. Survey articles and communications of the international conference held in Friedrichsroda, Germany, September 20-26, 1992, H.J. Schmeisser, H. Triebel, eds., vol. 133 of Teubner-Texte Math., Teubner, Stuttgart, Leipzig, 1993, pp. 9-126.

2. L. CONSIGLIERI, M.M.C. MUNIZ, Existence of solutions for a free boundary problem in the thermoelectrical modelling of an aluminium electrolytic cell, Eur. J. Appl. Math., 14 (2003), pp. 201-216.

3. M. DAUGE, Neumann and mixed problems on curvilinear polyhedra, Integral Equations Oper. Theory, 15 (1992), pp. 227-261.

4. P. DEGOND, S. GÉNIEYS, A. JÜNGEL, A steady-state system in non-equilibrium thermodynamics including thermal and electrical effects, Math. Methods Appl. Sci., 21 (1998), pp. 1399-1413.

5. J. ELSCHNER, G. SCHMIDT, Diffraction in periodic structures and optimal design of binary gratings. I. Direct problems and gradient formulas, Math. Methods Appl. Sci., 21 (1998), pp. 1297-1342.

6. J. ELSCHNER, R. HINDER, F. PENZEL, G. SCHMIDT, Existence, uniqueness and regularity for solutions of the conical diffraction problem, Math. Models Methods Appl. Sci. 10, (2000), pp. 317-341.

7. P. FABRIE, T. GALLOUËT, Modeling wells in porous media flow, Math. Models Methods Appl. Sci., 10 (2000), pp. 673-709.

8. H. GAJEWSKI, Analysis und Numerik von Ladungstransport in Halbleitern (Analysis and numerics of carrier transport in semiconductors), Mitt. Ges. Angew. Math. Mech., 16 (1993), pp. 35-57.

9. H. GAJEWSKI, K. GRÖGER, Initial boundary value problems modelling heterogeneous semiconductor devices, in: Surveys on Analysis, Geometry and Math. Phys., vol. 117 of Teubner-Texte Math., Teubner, Leipzig, 1990, pp. 4-53.

10. H. GAJEWSKI, H.-CHR. KAISER, H. LANGMACH, R. NÜRNBERG, R.H.  RICHTER, Mathematical modeling and numerical simulation of semiconductor detectors, in: Mathematics -- Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jäger, H.J. Krebs, eds., Springer, Berlin, Heidelberg, 2003, pp. 355-364.

11. A. GLITZKY, R. HÜNLICH, Global estimates and asymptotics for electro-reaction-diffusion systems in heterostructures, Appl. Anal., 66 (1997), pp. 205-226.

12. N. GRACHEV, V. MAZ'YA, On a contact problem for the Laplace operator in the exterior of the boundary of a dihedral angle, Math. Nachr., 151 (1991), pp. 207-231.

13. J.A. GRIEPENTROG, K. GRÖGER, H.-CHR. KAISER, J. REHBERG, Interpolation for function spaces related to mixed boundary value problems, Math. Nachr., 241 (2002), pp. 110-120.

14. J.A. GRIEPENTROG, H.-CHR. KAISER, J. REHBERG, Heat kernel and resolvent properties for second order elliptic differential operators with general boundary conditions on L^p, Adv. Math. Sci. Appl., 11 (2001), pp. 87-112.

15. K. GRÖGER, A W^{1, p}-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283 (1989), pp. 679-687.

16. D. HÖMBERG, A mathematical model for induction hardening including mechanical effects, Nonlinear Anal., Real World Appl., 5 (2004), pp. 55-90.

17. D. JERISON, C.E. KENIG, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), pp. 161-219.

18. A. JÜNGEL, Regularity and uniqueness of solutions to a parabolic system in nonequilibrium thermodynamics, Nonlinear Anal., Theory, Methods, Appl., 41 (2000), pp. 669-688.

19. H.-CHR. KAISER, H. NEIDHARDT, J. REHBERG, Classical solutions of quasilinear parabolic systems on two dimensional domains, Nonlinear Differ. Equ. Appl., in press.

20. R.B. KELLOG, On the Poisson equation with intersecting interfaces, Appl. Anal., 4 (1975), pp. 101-129.

21. V. KOZLOV, V. MAZ'YA, J. ROSSMANN, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, vol. 85 of Math. Surv. Monogr., American Mathematical Society (AMS), Providence, RI, 2001.

22. V. MAZ'YA, J. ELSCHNER, J. REHBERG, G. SCHMIDT, Solutions for quasilinear nonsmooth evolution systems in L^p, Arch. Ration. Mech. Anal., in press.

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