CC BY
86Probl. Anal. Issues Anal. Vol. 7 (25), Special Issue, 2018, pp. 101-112 101
DOI: 10.15393/j3.art.2018.5330
The paper is presented at the conference "Complex analysis and its applications" (COMAN 2018), Gelendzhik - Krasnodar, Russia, June 2-9, 2018.
UDC 517.95
A. G. LOSEV, E. A. MAZEPA
ON SOLVABILITY OF THE BOUNDARY VALUE PROBLEMS FOR THE INHOMOGENEOUS ELLIPTIC EQUATIONS ON NONCOMPACT RIEMANNIAN MANIFOLDS
Abstract. We study questions of existence and belonging to a given functional class of solutions of the inhomogeneous elliptic equations Au — c(x)u = g(x), where c(x) ^ 0, g(x) are Holder fuctions on a noncompact Riemannian manifold M without boundary. In this work we develop an approach to evaluation of solutions to boundary-value problems for linear and quasilinear equations of the elliptic type on arbitrary noncom-pact Riemannian manifolds. Our technique is essentially based on an approach from the papers by E. A. Mazepa and S. A. Ko-rol'kov connected with an introduction of equivalency classes of functions and representations. We investigate the relationship between the existence of solutions of this equation on M and outside some compact set B C M with the same growth "at infinity".
Key words: Riemannian manifold, nonhomogeneous elliptic equations, boundary-value problems
2010 Mathematical Subject Classification: 31C12
1. Introduction. This article is devoted to the investigation of the behavior of solutions of the inhomogeneous linear elliptic equation in relation to the geometry of the manifold in question. Such problems originate in the classification theory of noncompact Riemannian surfaces and manifolds (see [3]). For a noncompact Riemann surface, the well-known problem of the conformal type identification can be stated as follows: does a nontrivial positive superharmonic function exist on this surface?
© Petrozavodsk State University, 2018
Exactly this property served as a basis for the extension of the parabol-icity notion for arbitrary Riemannian manifolds. Namely, manifolds on which any lower bounded superharmonic function is constant are called parabolic manifolds.
Many questions of this kind fit into the pattern of a Liouville type theorem saying that the space of bounded solutions of some elliptic equation is trivial.
In works of a number of authors the conditions ensuring the validity of the Liouville property on noncompact Riemannian manifolds are adduced in terms of volume growth, or isoperimetric inequalities, and so on (see [3, 5,6,8,10]). However, the class of manifolds admitting nontrivial solutions of some elliptic equations is wide. For example, conditions ensuring the solvability of the Dirichlet problem with continuous boundary conditions "at infinity" for several noncompact manifolds has been found in many papers (see, e.g., [1,5,7,10,15]).
Notice that even the formulation of boundary-value problems for elliptic differential equations (in particular, the Dirichlet problem) on non-compact Riemannian manifolds and in unbounded domains of that manifolds can be problematic, since it is unclear how we should interpret the boundary data. Geometric compactification enables us sometimes to define them analogously the the classical statement of the Dirichlet problem in bounded domains of Rn (see, e.g., [1,10]).
In recent years, a large number of works were devoted to solvability of various boundary-value problems for harmonic functions, to solutions of stationary Schrodinger equation, for some other homogeneous linear, nonlinear and quasilinear elliptic equations. But studies of inhomogeneous elliptic equations are of a single nature [9,12,14].
In this article we study some questions of existence and belonging to given functional classes of solutions of the inhomogeneous elliptic equation
Lu = A u — c(x)u = g(x), (1)
where c(x), g(x) G C0'a(Q) for any subset Q CC M, 0 < a < 1 on a noncompact Riemannian manifold M without boundary, c(x) ^ 0.
Throughout the paper, we denote by Ck'a(Q) the subspace Ck(Q) consisting of all functions whose derivatives of order k are locally Holder continuous functions with index a, 0 < a < 1, k = 0, 1, 2... (see [2, pp. 57-59]).
In our research we use a new approach which is based on the consideration of the equivalence classes of functions on M. Previously, the
approach described below was used to study the solvability of boundary value problems for the Laplace-Beltrami equation, the Schrodinger equation and also for series of semilinear and quasilinear elliptic equations on arbitrary noncompact Riemannian manifolds (see, e.g., [7,13]).
The proof of the main results is based on the classical propositions of the theory of partial differential equations: the Maximum Principle, the Comparison and Uniqueness Theorems for solutions to linear elliptic differential equations. Their validity on precompact subsets of manifold M can be shown in just the same way as for bounded domains in Rn (see [2, pp. 39-40]).
2. Main concepts and auxiliaries. Let M be an arbitrary smooth connected noncompact Riemannian manifold without boundary and let {Bkbe an exhaustion of M, i.e., a sequence of precompact open subsets of M such that Bk с Bk+1 and M = U^Bfc. Throughout the sequel, we assume that the boundaries dBk are C1-smooth submanifolds.
Let /1 and /2 be arbitrary continuous functions on M.
Definition 1. [13] Say that f1 and /2 are equivalent on M and write /1 ~ /2 if for some exhaustion {Bkof M we have
lim sup |/i - /21 =0.
fc^TO M\Bk
It is easy to verify that the relation " ~ " is an equivalence which does not depend on the choice of the exhaustion of the manifold and so it partitions the set of all continuous functions on M into equivalence classes. Denote the equivalence class of a function / by [/].
Let B с M be an arbitrary connected compact subset and the boundary of B be a C 1-smooth submanifold. Assume that the interior of B is non-empty and B с Bk for all k.
Definition 2. We say that the boundary-value problem for equation (1) is solvable on M with boundary data from the class [/], if (1) has a solution u € [/] on M.
Definition 3. Let Ф(х) € C(dB) be any function continuous on dB. We say that the boundary-value problem for equation (1) is solvable on M \ B with boundary data (Ф, [/]) if (1) has a solution u(x) on M \ B such that
u € [/] and u|dB = ф|эв.
Note that if the manifold M has compact boundary or there is a natural geometric compactification of M (for example, on manifolds of negative
sectional curvature or spherically symmetric manifolds) which adds the boundary at infinity, then this approach naturally leads to the classical statement of the Dirichlet problem (see, for instance, [10]).
Now we formulate without proofs some auxiliary assertions. Detailed proofs of these statements can be found in [11,13,14].
The Comparison Principle. Let Lv ^ Lu on M \ B, v\dB ^ u\dB, v ~ u. Then v ^ u on M \ B. If Lv ^ Lu on M and v ~ u, then v ^ u on M.
The Uniqueness Theorem. Let Lv = g(x), Lu = g(x) on M \ B and v\dB = u\dB, v ~ u. Then w = u on M \ B.
Let Lv = g(x), Lu = g(x) on M and v ~ u. Then v = u on M.
Lemma 1. Suppose that G CC M is a precompact subset in M, a function u € C(G) n C2(G) satisfies the equation Lu = g on G, where g € C0(G), il := supp g and il CC G, c ^ 0 on G and c = 0 on il. Then
sup \u\ ^ sup \u\ + sup —.
G dG n c
Consider, together with equation (1), the homogeneous linear equation
Lu = Au — c(x)u = 0, (2)
which is the stationary Schrodinger equation.
Denote by vk the solution of equation (2) in Bk \ B that satisfies the conditions
vk\qb = 1, vk\dBk =0.
We can easily verify that the sequence vk is uniformly bounded on M \ B and so it is compact in the class C2(G) for every compact subset G C M \ B. Moreover, as k ^ <x> this sequence increases monotonically and converges on M \ B to a solution of equation (2)
v = lim vk, 0 < v ^ 1, v\dB = 1.
k^w
Also, note that the function v is independent of the choice of an exhaustion
{Bk 1^=1 (see, e.g., [7,13]).
Definition 4. [13] We call v the L-potential of the compact set B relative to M.
For the Laplace-Beltrami equation, the function v is the capacity potential of the compact set B relative to the manifold M (see [3]).
Definition 5. [11,13] Call the manifold M L-strict if for some compact set B C M there exists an L-potential v of B such that v € [0].
The L-strictness is shown to be independent of the choice of the compact set B in [11].
Remark 1. The connections between solvability of boundary-value and exterior boundary-value problems for linear and quasilinear homogeneous equations is investigated in details in [11,13].
Remark 2. In proving the main results, the asymptotic behavior of the solutions of Laplace-Beltrami and Schrodinger equations plays an important role. It is noted that the cases with c(x) = 0 and c(x) ^ 0 on the noncompact Riemannian manifold M are served by various theorems [3].
3. The main results for c(x) ^ 0.
Theorem 1. Let B C M be some connected compact subset such that c(x) > 0 on some neighborhood of B and the boundary-value problem for equation (1) is solvable with boundary data (A, [f]) on M \ B for any constant A. Then the boundary-value problem for equation (1) with boundary data from the class [f] is solvable on M too.
Proof. In what follows we assume that the subset B C M is chosen so that c(x) > 0 on some neighborhood B'' of B.
First, note that the condition of the theorem implies the existence on M \ B of a nontrivial capacity potential v € [0]. Let u0 be a solution of equation (1) on M \ B such that u0 € [f] and u\dB = 0\dB. Consider the function U0 € C2,a(M) such that U0 = u0 outside of B", U0 = 0 on the precompact B' CC B. Then LU0 = g0(x) on M, where the function g0(x) € Ca(M) and satisfies the following conditions: g0(x) =0 on the set B', g0(x) = g(x) outside of B'', g0(x) ^ g(x) on B'' \ B'.
Consider now the sequence of functions <k that are solutions of the problems
\L<k = g(x) in Bk,
\<k \dBk = u0\dBk
and the sequence of functions ^k = <k — U0. It is clear that ^k are solutions of the problems
jL^fc = g(x) - go(x) in Bfc,
|ÔBfc =°.
where the function g(x) — g0(x) G Ca(M) and satisfies the following conditions: g(x) — g0(x) = g(x) on the compact set B', g(x) — g0(x) = ° outside of B''. Thus, Q := supp{g(x) — g0(x)} is compact and Q C B''. By Lemma 1, for all k for x G Bk we have:
1,1/ 1,1, |g(x) — g0(x)| |g(x) — g0(x)|
—| < sup |—k| < sup — | + sup-—-= sup -
Bk dBk n c(x) n c(x)
which implies the uniform boundedness of the family of functions {—k}fc=i on M.
Hense, we obtain compactness of this family in the class C2(G) for an arbitrary compact subset G C M.
This in turn implies existence of the limit function — = limk^TO —k on M such that L— = g(x) — g0(x) on M.
Now we shall show that — G [0]. It is clear that — is a solution of the equation L— = 0 on M \ B''. Since dB'' is compact by continuity of the function —, there exists A = maxdB» | and we have
—A < —|SB„ < A
and also
—(A + 1) < —k U" < A + 1
for sufficiently large values k. Consider the functions
— = (A + 1) • v and — = —(A +1) • v
on M \ B'', where v is the L-potential of the compact set B'', v G [0]. The functions — and — are solutions to equation (1) and satisfy the conditions
-\дB„ = A +1, 0 < — < A +1, — G [0],
É|aB„ = —(A +1), — (A + 1) < — < 0, — G [0]. Then — < — on M \ B''. Since
L— = L—k = L— = 0,
±\dBk ^ ^k\8Bk < ^\8Bk
and
±\dB» ^ ^k \dB" < ^\9B"
on Bk \ B'', the Comparison Principle implies
^ < ^k < ^
on Bk \ B'' for sufficiently large values k. Passing to the limit as k ^ <x> we obtain ^ < ^ < Since ^ ~ ^ ~ 0 we have ^ € [0].
Finally, the existence of the function ^ = limk^TO r^k implies the existence of the limit function u = limk^TO <k such that Lu = g(x) on M and u ~ u0. The proof of Theorem 1 is over. □
Corollary 1. Let for any continuous function $(x) € C(dB) the boundary-value problem for equation (1) be solvable on M \ B with boundary data ($,[f]). Then the boundary-value problem for equation (1) with boundary data from the class [f] is solvable on M too.
Theorem 2. Let M be an L-strict manifold and the boundary-value problem for equation (1) is solvable on M with boundary data from the class [f]. Then for any continuous function $(x) € C(dB) the boundary-value problem for equation (1) is solvable on M \ B with boundary data ($, [f]).
Proof. We first prove that for every continuous function $ on dB there is a solution w to equation (2) on M\B, such that w\dB = $ and w € [0]. Consider the sequence of functions wk that are solutions to the boundary value problems:
Lwk = 0 in Bk \ B,
wk \bb = wk\dBk =
By the Maximum Principle, for every k we have
\wk \ < sup \wk \ = sup\$\,
d(Bk\B) dB
i.e., the sequence {wkis uniformly bounded on M and so it is compact in the class of twicely continuously differentiable functions on every compact subset of M. Let w(x) be a limit function. It is clear that
w\dB = $.
Put U = maxdB |$| and show that w G [0]. It is obvious that — (U + 1) < $ < U + 1, — (U +1) < w|dB < U + 1
and for every k
— (U + 1) < wk |ôB < U + 1.
Consider the functions w = —(U +1) • v and w2 = (U +1) • v on M\B, where v is the L-potential of the compact set B, v G [0]. The functions w1 and w2 are solutions to equation (1) and satisfy the conditions
ui|dB = —(U + 1), — (U +1) < wi < 0, wi G [0],
U2|dB = U +1, 0 < W2 < U +1, W2 G [0].
Then w1 ^ w2 on M \ B and, by the Comparison Principle, for all k we have
wi < Wk < «2
on Bk \ B. Taking the limit as k ^ to, we obtain wi ^ w ^ w2. Since wi ~ «2 ~ 0, we have w G [0].
Now, let w0 G [f] be a solution to the boundary value problem for equation (1) on M and $ be an arbitrary continuous function on dB. As has been shown above, there exists a solution w of equation (2) on M \ B such that w|dB = w0|dB — $ and w G [0]. Then the function w = w0 — w is a sought solution to the exterior boundary value problem for equation (1) on M \ B such that w G [f] and w|dB = $. □
4. The case of harmonic functions. Let B C M be an arbitrary connected compact subset, dB be an Ci-smooth submanifold. The fact that c(x) =0 on some compact set B C M was crucial in the proof of Theorem 1. However, the condition c = 0 on M does not violate Theorem 1 and Theorem 2 for the case of the Poisson equation
Aw = g(x), (3)
where g(x) G C°'a(Q) for any subset Q CC M, 0 < a < 1.
Similarly to the concept of L-strictness introduced above we can define the concept of A-strictness of the manifold M.
Definition 6. [13] A manifold M is called A-strict if there exists a nontrivial capacity potential v G [0] for some compact set B C M.
Note that the A-strictness of the manifold M implies the non-parabolicity of M (see, e.g., [3]). The following result holds.
Theorem 3. Let the boundary-value problem for equation (3) be solvable with boundary data (A, [f ]) on M \ B for any constant A. Then the boundary-value problem for equation (3) with boundary data from the class [f] is solvable on M too.
Proof. Let u0 be a solution of equation (3) on M\B such that u0 € [f ] and u0\dB = 0U. As in Theorem 1, consider the function U0 € C2'a(M) such that U0 = u0 outside of B'', U0 = 0 on the precompact B' CC B. Then AU0 = g0(x) on M, where the function g0(x) € C0'a(M) and satisfies the following conditions: g0(x) =0 on the set B', g0(x) = g(x) outside of B'', g0(x) £ g(x) on B'' \ B'.
Now consider the sequence of functions < k solving the problems
\A<k = g(x) in Bk, \<k \dBk = u0\dBk and the sequence of functions ^k = <k — U0. For these functions we have
\A^k = g(x) — g0(x) in Bk,
\dBk = 0
where the function g(x) — g0(x) € Ca(M) and satisfies the following conditions: g(x) — g0(x) = g(x) on the compact set B', g(x) — g0(x) = 0 outside of B''. Thus, l := supp{g(x) — g0(x)} is compact and l C B''.
Let Gk (x,y) be a Green's function in every Bk, i. e., the function which satisfies the conditions
AxGk (x,y) = —Sy (x), Gk (x,y) \xedBk = 0
for every y € Bk; here Sy(x) is the Dirac's ¿-function. Therefore, a Green representation
^k(x) = J Gk (x,y)(g(y) — g0(y))dy Bk
exists.
Note that the condition of the theorem implies existence of a nontrivial capacitive potential on the manifold M, and hence the non-parabolicity
of the manifold M [3]. Since the manifold M is non-parabolic, there is a finite limit of the Green's functions G(x,y) = limk^TO Gk (x,y) (see, e.g., [3]). The fact implies existence of the limit of the sequence {"k}. Let limk^TO " k = Then A— = g(x) — g0(x) on M. As in Theorem 1, we can prove that " G [0]. Hence, there is a limit function u = limk^TO of the sequence } and u satisfies u = " + U0, Au = g(x) on M and u ~ u0. The proof of the Theorem 3 is over. □
Corollary 1. Let the boundary-value problem for equation (3) be solvable with boundary data ($, [f ]) on M \ B for any continuous function $(x) G C(dB). Then the boundary-value problem for equation (3) with boundary data from the class [f ] is solvable on M too.
Theorem 4. Let M be a A-strict manifold and the boundary-value problem for equation (3) is solvable on M with boundary data from the class [f]. Then the boundary-value problem for equation (3) is solvable with boundary data ($, [f ]) on M \ B for any continuous function $(x) G C(dB).
Proof. The proof of this theorem coincides with the proof of the similar statement in Theorem 2. □
Remark 3. Note, that the given statements for the bounded continuous function f on M are proved in [14]. The exact conditions for the solvability of the Dirichlet problem for the Poisson equation on model manifolds are obtained in [9].
The obtained results can find application in the development of functional-analytic methods in the theory of elliptic equations on non-compact Riemannian manifolds. A more detailed description of these methods can be found, for example, in [3,4].
Acknowledgment. The author would like to acknowledge his students D.K. Navruzova, K.A. Bliznuk for their help in discussing the results and in preparing the article.
This work was supported by the Ministry of Science and Higher Education of the Russian Federation (government task 2.852.2017/4.6).
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Received May 29, 2018.
In revised form, August 28, 2018. Accepted August 31, 2018. Published online September 10, 2018.
A. G. Losev
Volgograd State University
100 Universitetsky pr., Volgograd 400062, Russia
E-mail: [email protected]
E. A. Mazepa
Volgograd State University
100 Universitetsky pr., Volgograd 400062, Russia
E-mail: [email protected]
