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# Uniqueness results for the Dirichlet problem for higher order elliptic equations in polyhedral angles

- Sara Monsurrò
^{1}Email author, - Ilia Tavkhelidze
^{2}and - Maria Transirico
^{1}

**2014**:232

https://doi.org/10.1186/s13661-014-0232-1

© Monsurrò et al.; licensee Springer. 2014

**Received:**23 June 2014**Accepted:**8 October 2014**Published:**4 November 2014

## Abstract

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.

## Keywords

- higher order elliptic equations
- Dirichlet problem
- discontinuous coefficients

## 1 Introduction

where ${\mathrm{\Delta}}^{m}$ denotes the polyharmonic operator of order *m*, Δ is the Laplace operator and ${\mathbb{R}}_{l}^{n}$ is a polyhedral angle of ${\mathbb{R}}^{n}$, defined in Section 2. We explicitly observe that for $l=0$ the above mentioned definition gives the half-space ${\mathbb{R}}_{+}^{n}$. We note that, due to the tools used in the proof, some restrictions on the dimension *n* of the space are required.

*m*:

where the discontinuous coefficients ${a}_{\alpha \beta}$ are bounded and measurable functions satisfying the uniform ellipticity condition.

Let us remark that if we take $\alpha =\beta $ and if the coefficients of the equation are constants ${a}_{\alpha \beta}(x)=\frac{m!}{\alpha !}$, then the left-hand side of the equation in (1.2) is exactly the polyharmonic operator ${\mathrm{\Delta}}^{m}$ in (1.1).

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 $m=1$ and in Section 5 we assume that $\alpha =\beta $. The main tool in our analysis is a generalization of the Hardy inequality proved by Kondrat’ev and Oleinik in [5] (see Section 3).

## 2 Notation

Throughout this work we use the following notation:

$n\in \mathbb{N}$ is the dimension of the considered space;

Greek letters denote *n*-dimensional multi-indices, for instance $\alpha =({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})$, where ${\alpha}_{i}\in \mathbb{N}\cup \{0\}$, $i=1,\dots ,n$;

$|\alpha |={\alpha}_{1}+\cdots +{\alpha}_{n}$ is the module of the multi-index *α*;

$\alpha !={\alpha}_{1}!\cdots {\alpha}_{n}!$ is the factorial of the multi-index *α*;

${\phi}_{,i}(x)=\frac{\partial \phi (x)}{\partial {x}_{i}}$, $i=1,\dots ,n$;

${D}_{i}^{{\alpha}_{i}}=\frac{{\partial}^{{\alpha}_{i}}}{{(\partial {x}_{i})}^{{\alpha}_{i}}}$, $i=1,\dots ,n$;

${D}^{\alpha}={D}_{1}^{{\alpha}_{1}}\cdots {D}_{n}^{{\alpha}_{n}}$;

for $\xi =({\xi}_{1},\dots ,{\xi}_{n})\in {\mathbb{R}}^{n}$ we set ${\xi}^{\alpha}={\xi}_{1}^{{\alpha}_{1}}\cdots {\xi}_{n}^{{\alpha}_{n}}$;

is the ‘polyhedral angle’ with vertex in the origin;

for $l=0$ the above definition gives the half-space ${\mathbb{R}}_{+}^{n}$;

for $\rho >0$ we denote by ${Q}_{\rho}=\{x\in {\mathbb{R}}_{l}^{n}:|x|<\rho \}$.

## 3 Setting of the problem

*m*, $m\in \mathbb{N}$, in certain unbounded domains of ${\mathbb{R}}^{n}$, $n>2$:

*i.e.*there exist two positive constants ${\lambda}_{1}$ and ${\lambda}_{2}$ such that for each nonzero vector $\xi \in {\mathbb{R}}^{n}$ one has

Let us mention that if we take $\alpha =\beta $ in (3.1) and if the coefficients of the equation are constants ${a}_{\alpha}(x)=\frac{m!}{\alpha !}$, then left-hand side of this equation is the polyharmonic operator ${\mathrm{\Delta}}^{m}$, where Δ denotes, as usual, the Laplace operator.

*u*and

*v*let us set

### Definition 3.1

*u*is a generalized solution of (3.1) in ${\mathbb{R}}_{l}^{n}$ with homogeneous Dirichlet boundary conditions if $u\in {W}^{m,2}({\mathbb{R}}_{l}^{n})$ and it satisfies the integral identity

for any $\rho >0$ and any function $v\in {W}_{0}^{m,2}({Q}_{\rho})$, where $f\in {L}^{2}({\mathbb{R}}_{l}^{n})$.

To prove our main results, consisting in two uniqueness theorems, we will essentially use the following generalized Hardy inequality.

### Lemma 3.2

(Generalized Hardy inequality)

*Let*$p>1$,

*j*,

*and*

*n*

*be such that*$j+n-p\ne 0$.

*Assume that for a sufficiently smooth function*

*g*

*the following condition is fulfilled in a cone*$V\subset {\mathbb{R}}^{n}$

*with vertex in the origin of coordinates*:

*where*$\u25bdg=(\frac{\partial g}{\partial {x}_{1}},\dots ,\frac{\partial g}{\partial {x}_{n}})$

*is the gradient of the function*

*g*.

*Then there exist two constants*$M,K>0$

*such that*

*where the constant* *K* *does not depend on the function* *g*. *If*, *in addition*, $g(0)=0$*then*$M=0$.

### Remark 3.3

with $0<{R}_{1}<{R}_{2}$, where ${V}_{R}$, $R>0$, 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 ${R}_{1}$ and ${R}_{2}$.

### Remark 3.4

As evidenced in many works about different variants of Hardy or Caffarelli-Kohn-Nirenberg type inequalities (see for instance [5], [12]–[16]), 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*.

## 4 Dirichlet problem for second order elliptic equations

Now we prove our first uniqueness result.

### Theorem 4.1

*Let*$n>2$. *Assume that* (3.2) *is satisfied*, *with*$m=1$. *If* *u* *is a generalized solution of problem* (4.1), *then*$u\equiv 0$*in*${\mathbb{R}}_{l}^{n}$.

### Proof

We note that in order to obtain a cut-off function $\mathrm{\Theta}(s)$ of the above mentioned type one can consider a classical mollifier and modify it suitably near to $s=1$ and $s=2$.

with ${K}_{2}={K}_{2}(n,{\lambda}_{1},{\lambda}_{2})$.

where the constant ${K}_{3}$ does not depend on the radius *R* and on the function *u* (see Remark 3.3).

*R*, we have, for any $P>0$,

This means that the function $u(x)$ is a constant and, according to the boundary condition in (4.1), this constant is zero. This concludes our proof. □

### Remark 4.2

Note that our proof do not provide any uniqueness result for $n=2$, 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*.

## 5 Dirichlet problem for higher order elliptic equations

*m*with homogeneous Dirichlet boundary conditions in the polyhedral angle ${\mathbb{R}}_{l}^{n}$, $l\in \{0,\dots ,n-1\}$:

### Theorem 5.1

*Let*$n>2m$*or*$n=2k+1$, *with*$k\in \mathbb{N}$. *Assume that* (3.2) *is satisfied*. *If* *u* *is a generalized solution of problem* (5.1), *then*$u(x)\equiv 0$*in*${\mathbb{R}}_{l}^{n}$.

### Proof

Let us use again the function ${\mathrm{\Theta}}_{R}$ introduced in the proof of Theorem 4.1.

where ${\mathrm{\Theta}}^{(i)}$ denotes the derivative of order *i* of the function Θ and ${P}_{|\alpha |}(x)$ is a polynomial of order $|\alpha |$.

where the constant ${K}_{\alpha}$ depends only on *α*.

with ${K}_{\alpha}^{\prime}={K}_{\alpha}^{\prime}(n,\alpha ,{\lambda}_{1},{\lambda}_{2},{K}_{\alpha})$ and ${K}_{\alpha}^{\u2033}={K}_{\alpha}^{\u2033}(n,\alpha ,{\lambda}_{1},{\lambda}_{2})$.

*m*. Thus, after an appropriate selection of ${\epsilon}_{\alpha}$, we get

where the constant $\tilde{K}$ is independent of the radius *R* and of the function *u*.

*R*, we have, for any $P>0$,

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 $u(x)\equiv 0$ in ${\mathbb{R}}_{l}^{n}$. □

### Remark 5.2

Clearly also in this case the repeated application of the Hardy inequality yields the restrictions $n>2m$ or $n=2k+1$, $k\in \mathbb{N}$, on the space dimension.

## Authors’ information

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).

## Declarations

## Authors’ Affiliations

## References

- 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 ${L}^{p}$-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 ${L}^{p}$ 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

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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.