CONTENTS Introduction Chapter I. General elliptic boundary-value problems § 1. The solubility of general elliptic boundary-value problems in domains with conic points § 2. The asymptotic behaviour of the solutions of a general boundary-value problem in the neighbourhood of a conic boundary point § 3. General boundary-value problems in non-smooth domains Chapter II. Boundary-value problems for the equations of mathematical physics in non-smooth domains § 1. Boundary-value problems for the system of elasticity theory § 2. Problems of hydrodynamics in domains with a non-smooth boundary § 3. The biharmonic equation Chapter III. Second-order elliptic equations in domains with a non-smooth boundary § 1. Boundary-value problems for second-order elliptic equations in an arbitrary domain § 2. Boundary-value problems in domains with isolated non-regular points on the boundary § 3. Second-order elliptic equations in domains with edges § 4. Boundary-value problems in domains that are diffeomorphic to a polyhedron Chapter IV. Parabolic and hyperbolic equations and systems in non-smooth domains § 1. Parabolic equations and systems in non-smooth domains § 2. Hyperbolic equations and systems in domains with singular points on the boundary References

https://iopscience.iop.org/article/10.1070/RM1983v038n02ABEH003470/pdf

Boundary-value problems for partial differential equations in non-smooth domains

Although the plane boundary value problem for the Laplacian with given Dirichlet data on one part Γ2 and given Neumann data on the remaining part Γ2 of the boundary is the simplest case of mixed boundary value problems, we present several applications in classical mathematical physics. Using Green's formula the problem is converted into a system of Fredholm integral equations for the yet unknown values of the solution u on Γ2 and the also desired values of the normal derivatie on Γ1. One of these equations has principal part of the second kind, whereas that one of the other is of the first kind. Since any improvement of constructive methods requires higher regularity of u but, on the other hand, grad u possesses singularities at the collision points Γ1 ∩ Γ2 even for C∞ data, u is decomposed into special singular terms and a regular rest. This is incorporated into the integral equations and the modified system is solved in appropriate Sobolev spaces. The solution of the system requires to solve a Fredholm equation of the first kind on the arc Γ2 providing an improvement of regularity for the smooth part of u. Since the integral equations form a strongly elliptic system of pseudodifferential operators, the Galerkin procedure converges. Using regular finite element functions on Γ1 and Γ2 augmented by the special singular functions we obtain optimal order of asymptotic convergence in the norm corresponding to the energy norm of u and also superconvergence as well as high orders in smoother norms if the given data are smooth (and not the solution).

On the integral equation method for the plane mixed boundary value problem of the Laplacian

Introduction. 1. Preliminaries. 2. Integral inequalities. 3. The Laplace operator. 4. Strong solutions of the Dirichlet problem for linear equations. 5. The Dirichlet problem for elliptic linear. divergent equations in a nonsmooth domain. 6. The Dirichlet problem for semilinear equations in a conical domain. 7. Strong solutions of the Dirichlet problem for nondivergence quasilinear equations. 8. Weak solutions of the Dirichlet problem for elliptic divergence form quasilinear equations. 9. The behavior of weak solutions to the boundary value problems for elliptic quasilinear equations with triple degeneration in a neighborhood of a boundary edge. 10. Sharp estimates of solutions to the Robin. boundary value problem for elliptic non divergence second order equations in a neighborhood of the conical point. Bibliography. Notation Index. Index.

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Elliptic boundary value problems of second order in piecewise smooth domains

The following investigation deals with surfaces governed by and extremal for a free energy functional which is quadratic in the principal curvatures. The associated Euler-Lagrange differential equations are derived, as are the corresponding intricate natural boundary conditions. Pertinent boundary value problems—without and with volume constraints—are formulated and discussed1 and existence proofs are provided for certain situations. The discussion opens the view onto an arena of rich mathematical problems which will also be of interest in engineering applications where the surfaces in question are utilized frequently as idealized models for the interfaces separating phases in real materials. 1. Let <5* = {x = \(u, v); (u, v) e P} be a differential geometric surface embedded in Euclidean space R3. The position vector x(w, v) = {x(u, v), y(u, v), z(u, v)) is defined over a prescribed parameter domain P—a connected set with or without given boundaries. We wish to consider this surface not so much as a geometric object but as an idealized model for the interfaces or middle surfaces occurring in real materials—open or closed lipid bilayers and surfactant films, thin elastic plates, 2 A etc. Thus, we associate with <5" a free energy O, per unit area, which incorporates surface tension but also reflects elastic properties. The latter involve the surface curvatures, and so <i> will have the form O = 0(«rx, k2) . Here kx = 1 //?, and k2 = 1 /R2 denote the principal curvatures of S". As a consequence, the total energy of S? becomes = <T(^) = J J <!>(*!, k2) dA . (1) Received August 6, 1991. 1991 Mathematics Subject Classification. Primary 35J35, 35J40, 35J60, 35J65, 49Q05, 49Q10, 53A10, 73C50; Secondary 31B30, 31G15, 35P15, 35Q35, 35R35, 53A05, 53C42, 76B45. 1 Some of the lengthier calculations are omitted. 2Realistically speaking, of course, the concept of sharp physical interfaces amenable to treatment with the tools of differential geometry is only an approximation, though often accurate and appropriate, to the actual narrow zones separating phases. For a discussion of the microstructure governing these zones in real materials, see, e.g. [1, 2, 11, 37, 39, 40, 63, 68]. ©1993 Brown University 363 364 JOHANNES C. C. NITSCHE (dA denotes the area element.) There may be further energy terms attributable to the boundary d5? of . Two universal structural conditions will be imposed on the integrand in (1): (1) 0(Kj , k2) is a symmetric function of its arguments. (2) if is definite in the following sense: There is a constant c > -oo , possibly negative, such that IP ) > c for every connected orientable surface -T of regularity class C2, with or without boundary. We allow the lower bound for i? to be negative to reflect the fact that the free surface energy of an interface need not be positive; however, disregarding possible extraneous information concerning the magnitude of physical constants or the limiting influence of boundary conditions in specific applications (see, e.g., Sec. 10 and (34) below), we shall insist on the universal condition C > —oo . It should be stressed that our condition of definiteness is mathematical in nature; it does not take into account the structural properties of the admissible comparison surfaces J7" on the molecular level; see footnote 2. Under mild regularity assumptions (for a polynomial , as a consequence of the fundamental theorem on symmetric polynomials), condition (1) above implies that can be written in the form 4>(Kr,, k2) = O(H, K), where H = (k{ + k,)/2 and K = k{k2 are the mean and Gaussian curvatures of 5?, respectively. Moreover, rather than working with general nonlinear integrands, it is consistent with standard engineering practice, at least as a first step, that we limit consideration to the inclusion of terms up to those of second order, so that <I> becomes <!>(//, K) — a0 + axH + a2H + a}K . The second condition (2) of definiteness imposes specific restrictions on the coefficients a0, , a}. In fact, an analysis similar to that given on p. 14 of [55] shows that O must have the form <P(H,K) = a + P(H-H0)2-yK, (2) with material constants (surface tension, elastic moduli) a, /?, y satisfying the conditions Q> 0, 0 <y<p, (lyH2 <a{p y). (3) It is the integrand (2), as well as the complex relations springing from it, which will be at the center of the discussion to follow. This integrand, or expressions 2 2 \ 2 closely resembling it—invariably including O = (/c, + k2) — 4H~ , <]> = (jc, — k2) = 4{H2 K), and O = k2 + k2 — AH2 — 2K—can be found in numerous sources from the earliest times (see, e.g., S. D. Poisson [59, pp. 221-225], [60, p. 321], S. Germain [21, pp. 12, 16], G. R. Kirchhoff [44, p. 63], F. Casorati [7, p. 109], A. E. H. Love [48, p. 130]) and has attracted renewed attention through the work of W. Helfrich [34] and fellow scientists (among others, [17, 28, 35, 37, 38, 58]; for further references also [54, Sec. 11]). The case <J> = 1 leads to the venerable theory of minimal surfaces (see [49, 51, 54]), and the case <J> = H2—to be sure, in all investigations published so far exclusively restricted to closed surfaces, embedded or immersed (i.e., surfaces with possible self-intersections)—has been of considerable recent interest to geometers; references can be found in [6, 70, 71, 73, 77] and [54, Sec. 11]. Generally speaking, there are today few mathematical treatments and no BOUNDARY VALUE PROBLEMS FOR VARIATIONAL INTEGRALS 365 general existence theorems for the various physical boundary value problems arising in connection with (1), (2). The "natural" (or "intrinsic", "spontaneous") curvature H0 in (2) can often be related to the initial state of the interface or to the fact that the two sides of a bilayer may be chemically different; see, e.g., [34, 17, 28], The actual numerical values of the constants a, ft , y , and H0 are of secondary importance at the moment. In many applications, a is larger by orders of magnitude than ft and y, a fact which often lets us approach the interface 5? as the perturbation of an interface governed by the energy functional ff dA based on the integrand O = 1 , i.e., a minimal surface (or a surface of constant mean curvature), depending on /? and y as control parameters. In other cases, for instance for menisci and vesicles with a size in the range of a few hundred A, the influence of a and that of /?, y become comparable. 2 3 The case a = 0, H0 = 0, p = y > 0, i.e., O = H K, is special. Here the associated energy W has the value zero for all spheres and spherical caps. Thus there may exist continua of solution surfaces for our Problem 1 below, and specific solution surfaces will have to be characterized by their area or by the volume they enclose. 2. One observation: The inclusion of terms of third order, that is, the addition of a4H} + a5HK to the expression 0(//, K) in (2) above, leads to an indefinite energy functional unless aA = a5 = 0. This can be seen as follows. For thin cylinders for which K = 0, the term /H3 dA can be made dominatingly negative so that the coefficient a4 must be zero. As for the other part, consider as comparison surface the torus = {x = (a + b cos u) cosv , y = (a + bcosu) sinw , z = ±6 sin u; 0 <u,v< 2nj, (4) where 0 <b < a . Set k = b/a. For this torus, 1 a + 2b + cos u dA = b(a + 6 cos u) dudv, H = ± 1 cos u 2b a + bcosu ' K = b a + bcosu ' so that JJ dA = 4n2ah, jj HdA = ±2n2a, J J KdA — 0, ff H2 dA = -iL== , ff HK dA = ± -=—_ n" JJr Av 1 X J J? ay 1 A (1 + vl A ) Obviously, the energy If (J7") can be made as negative as desired for a suitable choice of the radii a and b . Thus we must also have a5 = 0. It follows that the next step toward greater generality in the selection of the integrand 0(k, , k2) would have to include fourth powers of the principal curvatures. 3The combination (H2 K) dA is invariant under conformal transformations of i?3 ; see [8, 74]. The same is true for the differential equation (25); see [9], 366 JOHANNES C. C. NITSCHE On the phenomenological level, there are suggestions of such integrands; see, e.g., [3, 17, 37, 38]. On the mathematical level, the attending problems are formidable. 3. The static equilibrium shape of our interface 5? is determined by the condition that S' be energy minimizing or, more generally and less restrictive, that S? be an extremal for the energy functional % , i.e., S? must be a solution of the variational problem 8W = 0. This variational problem leads not only to the fundamental Euler-Lagrange differential equation, but also to certain specific intrinsic, or natural, boundary conditions. For the actual determination of these, we have to investigate the change of the functional f associated with a variation of the surface 5?. Given that the present discussion is not restricted to closed shapes (vesicles), as for instance the walls bounding blood cells, but includes surfaces with boundaries, fixed or free, the variations to be considered here cannot remain restricted to the special normal variations customarily employed in the literature. The surface S? will be regarded as the member of a one-parameter family of surfaces S"(t) = {x = x(u, v ; t); (u, v) e P), |f| < t0 , such that x(m , v ; 0) = x(u, v). We set y(u, v) — xt(u, v ; 0); then x{u ,v\t) = x(u, v) + ty(u, v) + 0(t2). A subsequent stability analysis would involve a discussion of the second variation S % and requires inclusion of the next term x[t(u, v ; 0) in the expansion for x(u ,v\t). Note that the variation y(w, v) has a component y—(y-X)X tangent to 5? and a component y [u ,v)' X{u, v) normal to S". Here X(u, v) — (X(u, v), Y(u, v), Z(u, v)) is the unit normal vector. For any functional & = &(S?), we shall use the standard abbreviation A rather lengthy computation leads to the following variation formulae: Sff dA = -2 [f H(yX)dAf (y,X,dx), (7) J J 5* J J5* JdS" S JJ *¥{H)dA= J J {AV H + 2{2H2 K)^ H 4H^}{y-X) dA -1 Jd2 JdS/ I dn dn ds,

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Boundary value problems for variational integrals involving surface curvatures

Publisher Summary This chapter discusses discretization of semiconductor device problems (I). It is possible to view classical or semiclassical modeling of transport in semiconductors as a hierarchical structure, sweeping from the Boltzmann Transport Equation (BTE) down to the Drift-Diffusion (DD) model, passing through all the systems derived from the moments of the BTE with respect to increasing powers of the carrier group velocity. The Hydrodynamic (HD) model and its inertial limit, that is a type of Energy-Transport (ET) model, are examples, among others, of these intermediate steps. From a mathematical viewpoint, when appropriate scalings are employed for the quantities appearing in the DD system, one finds that the conservation part of the system is convection dominated. In particular, there is a scaling, which makes the order of magnitude of the electron and hole concentrations equal to one. Looking at the structure of the Gummel map, one can recognize a nonlinear block Gauss–Seidel iteration that can be subdivided into two main loops: (1) a nonlinear inner iteration for solving the semilinear Poisson equation, and (2) two iterations for solving the continuity equations.

Neil M. Wigley

Papers

Mixed boundary value problems in plane domains with corners