- Open Access
Uniqueness results for the Dirichlet problem for higher order elliptic equations in polyhedral angles
© Monsurrò et al.; licensee Springer. 2014
- Received: 23 June 2014
- Accepted: 8 October 2014
- Published: 4 November 2014
We consider the Dirichlet boundary value problem for higher order elliptic equations in divergence form with discontinuous coefficients in polyhedral angles. Some uniqueness results are proved.
MSC: 35J30, 35J40.
- higher order elliptic equations
- Dirichlet problem
- discontinuous coefficients
where denotes the polyharmonic operator of order m, Δ is the Laplace operator and is a polyhedral angle of , defined in Section 2. We explicitly observe that for the above mentioned definition gives the half-space . We note that, due to the tools used in the proof, some restrictions on the dimension n of the space are required.
where the discontinuous coefficients are bounded and measurable functions satisfying the uniform ellipticity condition.
Our main results consist in two uniqueness theorems obtained for some particular cases of problem (1.2). More precisely, in Section 4 we consider problem (1.2) in the case and in Section 5 we assume that . The main tool in our analysis is a generalization of the Hardy inequality proved by Kondrat’ev and Oleinik in  (see Section 3).
Throughout this work we use the following notation:
is the dimension of the considered space;
Greek letters denote n-dimensional multi-indices, for instance , where , ;
is the module of the multi-index α;
is the factorial of the multi-index α;
for we set ;
is the ‘polyhedral angle’ with vertex in the origin;
for the above definition gives the half-space ;
for we denote by .
Let us mention that if we take in (3.1) and if the coefficients of the equation are constants , then left-hand side of this equation is the polyharmonic operator , where Δ denotes, as usual, the Laplace operator.
for any and any function , where .
To prove our main results, consisting in two uniqueness theorems, we will essentially use the following generalized Hardy inequality.
(Generalized Hardy inequality)
where the constant K does not depend on the function g. If, in addition, then.
with , where , , denotes the intersection between the cone V and the open ball of center in the origin and radius R.
This result can be deduced by the proof of Lemma 3.2, with slight modifications. We point out that in this proof it is also well rendered that the constant K does not depend on and .
As evidenced in many works about different variants of Hardy or Caffarelli-Kohn-Nirenberg type inequalities (see for instance , –), there are always very important restrictions on the dimension of the space n, the order of ‘singularity’ j and the order of the integral norm p.
Now we prove our first uniqueness result.
We note that in order to obtain a cut-off function of the above mentioned type one can consider a classical mollifier and modify it suitably near to and .
where the constant does not depend on the radius R and on the function u (see Remark 3.3).
This means that the function is a constant and, according to the boundary condition in (4.1), this constant is zero. This concludes our proof. □
Note that our proof do not provide any uniqueness result for , since in this case the generalized Hardy inequality in Lemma 3.2 does not apply, as a consequence of our choice of p and j.
Let us use again the function introduced in the proof of Theorem 4.1.
where denotes the derivative of order i of the function Θ and is a polynomial of order .
where the constant depends only on α.
with and .
where the constant is independent of the radius R and of the function u.
Therefore, the partial derivatives of any order of the solution are equal to zero, thus, as a consequence of the boundary conditions in (5.1), we deduce that in . □
Clearly also in this case the repeated application of the Hardy inequality yields the restrictions or , , on the space dimension.
Sara Monsurrò and Maria Transirico are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
- Sobolev SL: Some Applications of Functional Analysis in Mathematical Physics. Izdat. Leningrad. Gos. Univ., Leningrad; 1950.Google Scholar
- Vekua, IN: On metaharmonic functions. Lect. Notes TICMI 14, 62 pp. (2013). Translated from the 1943 Russian original Trav. Inst. Math. Tbilissi 12, 105-174 (1943)Google Scholar
- Vekua IN: New Methods for Solving Elliptic Equations. OGIZ, Moscow; 1948.Google Scholar
- Cassisa C, Ricci P, Tavkhelidze I: Analogue of Saint-Venant’s principle for the one special type 4-th order elliptic equation and its applications. Appl. Math. Inform. 1999, 4: 11-29.MathSciNetGoogle Scholar
- Kondrat’ev VA, Oleinik OA: Boundary value problems for the system of elasticity theory in unbounded domains. Korn’s inequalities. Usp. Mat. Nauk 1988, 43(5):55-98. (in Russian)MathSciNetGoogle Scholar
- Kozlov V, Maz’ya V: Asymptotics of a singular solution to the Dirichlet problem for an elliptic equation with discontinuous coefficients near the boundary. In Function Spaces, Differential Operators and Nonlinear Analysis. Birkhäuser, Basel; 2003:75-115. (Teistungen, 2001) 10.1007/978-3-0348-8035-0_5View ArticleGoogle Scholar
- Monsurrò S, Transirico M:A -estimate for weak solutions of elliptic equations. Abstr. Appl. Anal. 2012., 2012: 10.1155/2012/376179Google Scholar
- Monsurrò S, Transirico M: Dirichlet problem for divergence form elliptic equations with discontinuous coefficients. Bound. Value Probl. 2012., 2012: 10.1186/1687-2770-2012-67Google Scholar
- Monsurrò S, Transirico M:A priori bounds in for solutions of elliptic equations in divergence form. Bull. Sci. Math. 2013, 137: 851-866. 10.1016/j.bulsci.2013.02.002MathSciNetView ArticleGoogle Scholar
- Tavkhelidze I: On some properties of solutions of polyharmonic equation in polyhedral angles. Georgian Math. J. 2007, 14: 565-580.MathSciNetGoogle Scholar
- Gazzola F, Grunau H, Sweers G: Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains. Springer, Berlin; 2010.Google Scholar
- Bojarski B, Hajlasz P: Pointwise inequalities for Sobolev functions and some applications. Stud. Math. 1993, 106: 77-92.MathSciNetGoogle Scholar
- Brezis H, Marcus M: Hardy’s inequalities revisited. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 1997, 25: 217-237. Dedicated to Ennio De GiorgiMathSciNetGoogle Scholar
- Catrina F, Wang ZQ: On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Commun. Pure Appl. Math. 2001, 54: 229-258. 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-IMathSciNetView ArticleGoogle Scholar
- Dávila J, Dupaigne L: Hardy-type inequalities. J. Eur. Math. Soc. 2004, 6: 335-365. 10.4171/JEMS/12View ArticleGoogle Scholar
- Filippas S, Tertikas A, Tidblom J: On the structure of Hardy-Sobolev-Maz’ya inequalities. J. Eur. Math. Soc. 2009, 11: 1165-1185. 10.4171/JEMS/178MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.