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Optimal regularity for elliptic operators with nonsmooth data

Collaborator: J. Elschner (FG 4), H.-Chr. Kaiser (FG 1), J. Rehberg (FG 1), G. Schmidt

 (FG 4)

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., 2002, p. 39 et sqq., and 2003, p. 178 et sqq.).

In view of the applications which we have in mind, one is nearly always confronted with nonsmooth domains, discontinuous coefficients and/or mixed boundary conditions. In particular, in simulation of the current flow in real semiconductor devices---a project area of WIAS---the latter is unavoidable, or the model is meaningless, [18]. As is well known, the usual regularity properties which hold in case of smooth constellations then break down. So, one has to look for suitable substitutes. It turns out that for the successful treatment of such equations it is of great use to have optimal regularity results for the corresponding elliptic operators - $ \nabla$ . $ \mu$$ \nabla$ between W1, q$\scriptstyle \Gamma$ and W-1, q$\scriptstyle \Gamma$ (with homogeneous Dirichlet boundary conditions on $ \partial$$ \Omega$ $ \setminus$ $ \Gamma$) at hand where q is larger than the space dimension. In [12], Gröger proved that q may be taken always larger than 2 under very general conditions on the domain, the coefficients, and the Neumann boundary part. This result is very useful in many spatially two-dimensional applications, see [2, 4, 9, 11, 14]. One cannot expect q > 3 in this general setting (see [17] and the Example below). But the necessity grows to study not only two-dimensional problems (mostly as cuts of the original three-dimensional ones) but the three-dimensional models themselves; this means that one needs q > 3. This is true in semiconductor device modeling (see [10], [15]) but also in many other application areas. Thus, we intend to define a class of three-dimensional domains $ \Omega$, (possibly discontinuous) coefficient functions $ \mu$, and Neumann boundary parts $ \Gamma$ such that

$\displaystyle \nabla$ . $\displaystyle \mu$$\displaystyle \nabla$ : W$\scriptstyle \Gamma$1, q($\displaystyle \Omega$) $\displaystyle \mapsto$ W$\scriptstyle \Gamma$-1, q($\displaystyle \Omega$) (1)
is a topological isomorphism for a q > 3 (and, hence, by interpolation and duality, for all $ \tilde{q}$ $ \in$ [q', q]). Let us emphasize that our principal aim is to generate a setting where many situations of practical relevance, subject to research at the Weierstrass Institute and elsewhere, are included.

For some nonsmooth situations one has this already at hand: the pure Dirichlet and Neumann Laplacian on Lipschitz domains was considered in [13] and [19], respectively. Concerning mixed boundary conditions, M. Dauge proved in [3] the following result:

Proposition 1. Let $ \Omega$ be a convex polyhedron and $ \Gamma$ be the Neumann boundary part of $ \partial$$ \Omega$. If $ \bar{\Gamma}$ $ \cap$ ($ \partial$$ \Omega$ $ \setminus$ $ \Gamma$) is a connected curve which consists of finitely many line segments, then

- $\displaystyle \Delta$ : W1, q$\scriptstyle \Gamma$($\displaystyle \Omega$) $\displaystyle \longmapsto$ W-1, q$\scriptstyle \Gamma$($\displaystyle \Omega$) (2)
is a topological isomorphism for some q > 3.

Let us emphasize that in particular the suppositions are fulfilled if $ \Omega$ is the unit cube and $ \Gamma$ is one half of its ground plate.

While in the afore-mentioned cases, the coefficients are constant, we proved in a joint paper [16] with V. Maz'ya the following result:

Theorem 1. Let $ \Omega$ be a Lipschitzian polyhedron which splits up into finitely many subpolyhedra. Let $ \mu$ be a coefficient function on $ \Omega$ which takes its values in the set of real, symmetric, positive definite 3 x 3 matrices and is constant on any subpolyhedron.

For any edge E and $ \bf x$ $ \in$ E take an orthogonal transformation $ \mathcal {O}$E which transforms this edge into a line which is parallel to the z-axis. Consider the resulting transformed coefficient matrix function $ \mu_{{E,{\bf x}}}^{}$ under the mapping $ \bf y$ $ \mapsto$ $ \mathcal {O}$E$ \bf y$ - $ \mathcal {O}$E$ \bf x$. Denote by $ \hat{\mu}$ the upper left 2 x 2 matrix function, restricted to the intersection of the x-y plane with $ \mathcal {O}$E($ \Omega$ - $ \bf x$). By passing to polar coordinates (r,$ \theta$), multiplying by r2, and taking the Mellin transform with respect to r, one obtains from the corresponding two-dimensional boundary value problem the following one-dimensional problem with parameter $ \lambda$ on an interval of angles

$\displaystyle \Pi$($\displaystyle \lambda$)$\displaystyle \tilde{u}$$\displaystyle \;\stackrel{{\mathrm{def}}}{{=}}\;$ - $\displaystyle \partial_{\theta}^{}$(b2$\displaystyle \partial_{\theta}^{}$$\displaystyle \tilde{u}$) - $\displaystyle \lambda$$\displaystyle \partial_{\theta}^{}$(b1$\displaystyle \tilde{u}$) - $\displaystyle \lambda$b1$\displaystyle \partial_{\theta}^{}$$\displaystyle \tilde{u}$ - $\displaystyle \lambda^{2}_{}$b0$\displaystyle \tilde{u}$ = $\displaystyle \tilde{g}$  , (3)
supplemented by transmission conditions (see [16] for details).

If for every edge E, the possible $ \lambda$s, satisfying (3) with right-hand side zero and the transmission conditions, have real parts not contained in the interval ]0, 1/3], then there is a q > 3 such that

- $\displaystyle \nabla$ . $\displaystyle \mu$$\displaystyle \nabla$ : W01, q($\displaystyle \Omega$) $\displaystyle \longmapsto$ W-1, q

is a topological isomorphism.

Unfortunately, for each edge the set of occurring $ \lambda$s is identical with the set of zeros of a transcendental equation which is not at all easy to discuss in general. Nevertheless, we succeeded in proving the following result which covers completely the case of polygonal, layered structures:

Theorem 2 (to be published in [7]). Assume that the following conditions are fulfilled: $ \Omega$ is a three-dimensional Lipschitzian polyhedron. There are hyperplanes $ \mathcal {H}$1...$ \mathcal {H}$n in R3 intersecting (within $ \bar{\Omega}$) with each other at most in a vertex of the polyhedron such that the coefficient function $ \mu$ is a constant real symmetric positive definite 3 x 3 matrix on each of the connected components of $ \Omega$ $ \setminus$ $ \cup_{{l=1}}^{n}$ $ \mathcal {H}$l. Moreover, for every edge on the boundary, induced by a hetero-interface $ \mathcal {H}$l, the angles between the outer boundary plane and the hetero-interface do not exceed $ \pi$ and at most one of them may equal $ \pi$. Then the operator - $ \nabla$ . $ \mu$$ \nabla$ provides a topological isomorphism between W01, q($ \Omega$) and W-1, q($ \Omega$) for all q $ \in$ [2, 4].

In the case of nonsmooth interfaces, q exceeds 2 only arbitrarily little, as the following instructive counterexample by J. Elschner shows.

Example (to be published in [7]). Consider the following coefficient function $ \rho$ on R2:

$\displaystyle \rho$(x, y) = $\displaystyle \bigg($$\displaystyle \begin{array}{cc} 1 & 0\\ 0 & t^2 \end{array}$$\displaystyle \bigg)$if  x, y > 0        and        $\displaystyle \bigg($$\displaystyle \begin{array}{cc} t & 0\\ 0 & t \end{array}$$\displaystyle \bigg)$elsewhere onR2

(t positive) and, correspondingly, the following elliptic problem on R2:

$\displaystyle \nabla$ . $\displaystyle \rho$$\displaystyle \nabla$w = 0, w $\displaystyle \in$ W1, 2loc(R2). (4)
Following again [16], one evaluates the $ \lambda$s with the smallest positive real part as

$\displaystyle \lambda$ = $\displaystyle {\frac{{8 \pi^2}}{{4\pi^2+\ln^2t}}}$$\displaystyle \pm$i$\displaystyle {\frac{{4\pi \ln t}}{{4\pi^2+\ln^2t}}}$. (5)
If t $ \mapsto$ $ \infty$, then the real part of $ \lambda$ converges to 0. This provides a local solution of (4) which behaves like (x2 + y2)$\scriptstyle \lambda$/2 in a neighborhood of 0 $ \in$ R2. Tending with t to $ \infty$, these solutions lack any common (local) integrability of order larger than 2 for their first-order derivatives.

Clearly, the example is not restricted to two dimensions, namely, one can add arbitrarily many dimensions by extending the solution constantly in these directions---at least in a neighborhood of zero.

However, in case of an inner C1 interface, we proved the following result which generalizes the linear regularity result of [1].

Theorem 3 (to be published in [8]). Let $ \Omega$ be a Lipschitz domain and $ \Omega_{\circ}^{}$ $ \subset$ $ \Omega$ be a C1 domain which does not touch the boundary of $ \Omega$. Suppose that the coefficient function $ \mu$ is uniformly continuous on $ \Omega$ $ \setminus$ $ \bar{\Omega}_{\circ}^{}$ and on $ \Omega_{\circ}^{}$. Then there is a q > 3 such that

- $\displaystyle \nabla$ . $\displaystyle \mu$$\displaystyle \nabla$ : W1, q0 $\displaystyle \longmapsto$ W-1, q

is a topological isomorphism.

Further, from Proposition 1 one may conclude by localization techniques and perturbation arguments the following

Theorem 4 (to be published in [8]). Let $ \mathcal {Q}$ $ \subset$ R3 denote the unit cube and $ \Upsilon$ one half of its ground plate. Assume that the coefficient function $ \mu$ is uniformly continuous. For any $ \bf x$ $ \in$ $ \bar{\Gamma}$ $ \cap$ ($ \partial$$ \Omega$ $ \setminus$ $ \Gamma$) there is an open neighborhood $ \mathcal {U}$$\scriptstyle \bf x$ in R3 and a homeomorphism $ \Psi_{{\bf x}}^{}$ from $ \mathcal {U}$$\scriptstyle \bf x$ onto an open set $ \mathcal {V}$$\scriptstyle \bf x$ which satisfy $ \Psi$($ \mathcal {U}$$\scriptstyle \bf x$ $ \cap$ ($ \Omega$ $ \cup$ $ \Gamma$) = $ \mathcal {Q}$ $ \cup$ $ \Upsilon$ and $ \Psi$($ \bf x$) = 0. Moreover, every $ \Psi_{{\bf x}}^{}$ and its inverse are continuously differentiable and the derivative is uniformly continuous on $ \mathcal {U}$$\scriptstyle \bf x$. Then there is a q > 3 such that - $ \nabla$ . $ \mu$$ \nabla$ is a topological isomorphism between W1, q$\scriptstyle \Gamma$ and W-1, q$\scriptstyle \Gamma$.

Finally, one can define a global setting which includes all the above-mentioned situations in a certain sense as its local constituents, see [8].

References:

  1. L.A. CAFFARELLI, I. PERAL, On W1, p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51 (1998), pp. 1-21.

  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 Operator 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. J. ELSCHNER, H.-CHR. KAISER, J. REHBERG, G. SCHMIDT, W1, q regularity results for elliptic transmission problems on heterogeneous polyhedra, in preparation.

  8.          , Optimal regularity for elliptic operators occurring in real-world problems, in preparation.

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

  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-Verlag, Berlin, Heidelberg, 2003, pp. 355-364.

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  12. K. GRÖGER, A W1, p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283 (1989), pp. 679-687.

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

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

  15. G. LUTZ, R.H. RICHTER, L. STRÜDER, DEPMOS-Arrays for X-ray imaging, in: XRAY Optics, Instruments and Missions III, J. Trümper, B. Aschenbach, eds., Proceedings of SPIE, 4012 (2000), pp. 249-256.

  16. V. MAZ'YA, J. ELSCHNER, J. REHBERG, G. SCHMIDT, Solutions for quasilinear nonsmooth evolution systems in Lp, Arch. Ration. Mech. Anal., 171 (2004), pp. 219-262.

  17. N.G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17 (1963), pp. 189-206.

  18. S. SELBERHERR, Analysis and simulation of semiconductor devices, Springer-Verlag, Wien, New York, 1984.

  19. D. ZANGER, The inhomogeneous Neumann problem in Lipschitz domains, Comm. Partial Differential Equations, 25 (2000), pp. 1771-1808.


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2005-07-29