##
**Polyharmonic boundary value problems. Positivity preserving and nonlinear higher order elliptic equations in bounded domains.**
*(English)*
Zbl 1239.35002

Lecture Notes in Mathematics 1991. Berlin: Springer (ISBN 978-3-642-12244-6/pbk; 978-3-642-12245-3/ebook). xviii, 423 p. (2010).

The poly-Laplace or polyharmonic operator, an entire power of the Laplacian, naturally plays a central role among elliptic higher order partial differential operators. It is as well a model operator as it is the leading part of more general ones. While existence and regularity results are generally available for the basic case of second order, for higher order elliptic equations there is no maximum principle. This gap demands new methods to treat nonlinear higher order elliptic equations. The main tasks of the present Lecture Notes in Mathematics volume are nonlinear problems and positivity statements for higher order elliptic equations involving polyharmonic operators. In particular the biharmonic operator and semilinear operators related to it are investigated.

“Models of Higher Order” (Chapter 1) introduces the subject recalling linear boundary value problems for the plate equation and discussing basic and still open questions concerning qualitative properties of solutions of various linear boundary value problems and related eigenvalue problems such as positivity and almost positivity. Semilinear problems for bi- and polyharmonic operators are motivated with geometrical background. The Willmore functional, modeling the pure bending energy in terms of the squared mean curvature of an elastic surface, is discussed as a particular case of a model for thin elastic plates with full nonlinear differential geometric expression.

“Linear Problems” (Chapter 2) is a preparatory text for handling nonlinear ones. Higher order Sobolev spaces are presented, as well as relevant boundary conditions for polyharmonic problems. A wide class of boundary value problems is solved in suitable Hilbert spaces, and regularity results and a priory estimates in Schauder and \(L^p\) settings including maximum modulus estimates are discussed. Polyharmonic Green functions for the Dirichlet problem, in particular the Boggio formula for the unit ball are given. (Recently T. Sh. Kalmenov, B. D. Koshanov and M. Y. Nemchenko [Complex Var. Elliptic Equ. 53, No. 2, 177–183 (2008; Zbl 1157.31004)] have constructed an explicit, integral-free expression for this polyharmonic Green function for the ball.) Biharmonic problems in nonsmooth domains explain why the numerical approximation of solutions is problematic in general domains.

Chapter 3 is labeled “Eigenvalue Problems”. The Krein-Rutman approach is used to show the existence of a positive real eigenvalue, being the corresponding eigenfunctions also positive. This applies when the boundary value problem is “positivity preserving” even for a non-selfadjoint situation. For a selfadjoint case in Hilbert spaces a simple proof of positivity and simplicity of the first eigenvalue is provided by a dual cone approach. It is however shown that the connection between the positivity preserving property of the Dirichlet problem and the fixed sign property of the first eigenfunction may fail. Minimization problems are investigated for the biharmonic operator. In dimensions 2 and 3, among domains of equal measure, the ball achieves the minimum of the first Dirichlet eigenvalue. For the Steklov boundary value problem, for which the whole spectrum is investigated, no optimal shape of given measure is shown to exist. For the buckling load of a clamped plate the minimizing problem is discussed in connection with the Pólya-Szegö conjecture.

“Kernel Estimates” (Chapter 4) discusses the requirements for getting positivity and almost positivity statements for higher order boundary value problems. The polyharmonic Green functions, which, contrary to the harmonic case, in general change sign, and their derivatives, in particular the Poisson kernels, are estimated. At first the polyharmonic Green function for the Dirichlet problem is treated. Two-side estimates are attained in the case of the unit ball from the Boggio formula. A 3-G type result provides estimates for some terms in 3 Green functions. In case of boundary value conditions, which allow to reformulate the polyharmonic boundary value problem as a system of problems for the Poisson equation as e.g. the Navier, also called Riquier problem, iterated Green functions are useful. For this situation, general smooth bounded domains are admitted. (For certain plane domains such iterated polyharmonic Green functions were recently calculated explicitly, see e.g. [the reviewer and T. Vaitekhovich, Matematiche 63, No. 1, 139–154 (2008; Zbl 1193.31002)] and [the reviewer, Acta Math. Vietnam. 36, No. 2, 169–181 (2011; Zbl 1233.31002)]). In order to avoid technicalities, estimates are restricted to the biharmonic Green function. Also weighted estimates are provided and convergence of Green functions in domain approximation is discussed. Most of the material presented here is based on work of all three authors.

On the basis of the kernel estimates, positivity and lower order perturbations of polyharmonic boundary value problems are considered in Chapter 5. The positivity of the Dirichlet problem for \(2m\)-order elliptic equations with the polyharmonic operator as leading part and “small” lower order terms in the unit ball is proved. Without smallness restriction a local maximum principle for differential inequalities is developed also for arbitrary domains. The positivity preserving property is proved for the homogeneous Steklov boundary value problem for the bi-Poisson equation with suitable data and above a certain critical parameter. For the hinged plate in a convex domain this means upward pushing yields upward bending. For biharmonic functions the question is discussed if positive Dirichlet data imply positivity of the function. Very different situations occur.

Contrary to the harmonic case, a polyharmonic Green function is in general sign depending. For the biharmonic Green function with Dirichlet boundary data the relation of its positive and negative parts is discussed in Chapter 6, “Dominance of Positivity in Linear Equations”. It is shown that the negative part is relatively small. Uniform estimates are attained when the singularity approaches the boundary. While the planar case is quite involved and the related material with hints to the literature just sketched, the higher dimensional theory is explained in detail. A perturbation result of positivity is also presented. The biharmonic Green function is positive for the unit ball. This remains so under certain small perturbations of the ball. In dimension 2, perturbing the polyharmonic operator preserves positivity of the biharmonic Green function.

“Semilinear Problems” (Chapter 7) concerns the elliptic polyharmonic reaction-diffusion-type model equation mostly under Dirichlet boundary conditions not allowing decomposition in a system of second order Poisson equations with Dirichlet conditions. But also Navier and Steklov boundary conditions are looked at. The nonlinearity is assumed to be continuous and nondecreasing in order to prove existence and positivity results. Especially superlinear ones of the kind \[ f(u)=\lambda u+|u|^{p-1}u, \;\;\lambda\in \mathbb{R}, \;\;1<p, \] are investigated. Kernel estimates and monotonicity properties of the polyharmonic Green function are used to prove radial symmetry of positive solutions under Dirichlet conditions. Existence and nonexistence results for nontrivial solutions to the Dirichlet problem depend essentially on the value of the exponent \(p\). Assuming the space dimension exceeds the order of the equation, the sub-, super-, and critical case itself are studied. Much attention is paid to the geometrically relevant and complicated critical case. Here existence results are seen to depend strongly on the space dimension, the subcritical perturbations, and the geometry of the domain. For the supercritical equation only the bi-Laplace operator is considered, while the subcritical case turns out as not problematic. The critical growth equation is also studied under Navier and under Steklov boundary conditions. Here different phenomena occur.

Chapter 8, ‘’Willmore Surfaces of Revolution” presents an existence result for a priori bounded classical solutions to the Dirichlet problem for Willmore surfaces.

That the authors are experienced researchers on the topic of the volume becomes evident from the excellent, clear and well motivated presentation. Reading the closing sections “Biographical Notes” of chapters 2 to 8, together with the extensive list of 423 references, the engagement of each of the 3 authors becomes obvious. Whoever is interested in an exciting subject of the modern theory of higher order elliptic equations is recommended to study this exposition.

“Models of Higher Order” (Chapter 1) introduces the subject recalling linear boundary value problems for the plate equation and discussing basic and still open questions concerning qualitative properties of solutions of various linear boundary value problems and related eigenvalue problems such as positivity and almost positivity. Semilinear problems for bi- and polyharmonic operators are motivated with geometrical background. The Willmore functional, modeling the pure bending energy in terms of the squared mean curvature of an elastic surface, is discussed as a particular case of a model for thin elastic plates with full nonlinear differential geometric expression.

“Linear Problems” (Chapter 2) is a preparatory text for handling nonlinear ones. Higher order Sobolev spaces are presented, as well as relevant boundary conditions for polyharmonic problems. A wide class of boundary value problems is solved in suitable Hilbert spaces, and regularity results and a priory estimates in Schauder and \(L^p\) settings including maximum modulus estimates are discussed. Polyharmonic Green functions for the Dirichlet problem, in particular the Boggio formula for the unit ball are given. (Recently T. Sh. Kalmenov, B. D. Koshanov and M. Y. Nemchenko [Complex Var. Elliptic Equ. 53, No. 2, 177–183 (2008; Zbl 1157.31004)] have constructed an explicit, integral-free expression for this polyharmonic Green function for the ball.) Biharmonic problems in nonsmooth domains explain why the numerical approximation of solutions is problematic in general domains.

Chapter 3 is labeled “Eigenvalue Problems”. The Krein-Rutman approach is used to show the existence of a positive real eigenvalue, being the corresponding eigenfunctions also positive. This applies when the boundary value problem is “positivity preserving” even for a non-selfadjoint situation. For a selfadjoint case in Hilbert spaces a simple proof of positivity and simplicity of the first eigenvalue is provided by a dual cone approach. It is however shown that the connection between the positivity preserving property of the Dirichlet problem and the fixed sign property of the first eigenfunction may fail. Minimization problems are investigated for the biharmonic operator. In dimensions 2 and 3, among domains of equal measure, the ball achieves the minimum of the first Dirichlet eigenvalue. For the Steklov boundary value problem, for which the whole spectrum is investigated, no optimal shape of given measure is shown to exist. For the buckling load of a clamped plate the minimizing problem is discussed in connection with the Pólya-Szegö conjecture.

“Kernel Estimates” (Chapter 4) discusses the requirements for getting positivity and almost positivity statements for higher order boundary value problems. The polyharmonic Green functions, which, contrary to the harmonic case, in general change sign, and their derivatives, in particular the Poisson kernels, are estimated. At first the polyharmonic Green function for the Dirichlet problem is treated. Two-side estimates are attained in the case of the unit ball from the Boggio formula. A 3-G type result provides estimates for some terms in 3 Green functions. In case of boundary value conditions, which allow to reformulate the polyharmonic boundary value problem as a system of problems for the Poisson equation as e.g. the Navier, also called Riquier problem, iterated Green functions are useful. For this situation, general smooth bounded domains are admitted. (For certain plane domains such iterated polyharmonic Green functions were recently calculated explicitly, see e.g. [the reviewer and T. Vaitekhovich, Matematiche 63, No. 1, 139–154 (2008; Zbl 1193.31002)] and [the reviewer, Acta Math. Vietnam. 36, No. 2, 169–181 (2011; Zbl 1233.31002)]). In order to avoid technicalities, estimates are restricted to the biharmonic Green function. Also weighted estimates are provided and convergence of Green functions in domain approximation is discussed. Most of the material presented here is based on work of all three authors.

On the basis of the kernel estimates, positivity and lower order perturbations of polyharmonic boundary value problems are considered in Chapter 5. The positivity of the Dirichlet problem for \(2m\)-order elliptic equations with the polyharmonic operator as leading part and “small” lower order terms in the unit ball is proved. Without smallness restriction a local maximum principle for differential inequalities is developed also for arbitrary domains. The positivity preserving property is proved for the homogeneous Steklov boundary value problem for the bi-Poisson equation with suitable data and above a certain critical parameter. For the hinged plate in a convex domain this means upward pushing yields upward bending. For biharmonic functions the question is discussed if positive Dirichlet data imply positivity of the function. Very different situations occur.

Contrary to the harmonic case, a polyharmonic Green function is in general sign depending. For the biharmonic Green function with Dirichlet boundary data the relation of its positive and negative parts is discussed in Chapter 6, “Dominance of Positivity in Linear Equations”. It is shown that the negative part is relatively small. Uniform estimates are attained when the singularity approaches the boundary. While the planar case is quite involved and the related material with hints to the literature just sketched, the higher dimensional theory is explained in detail. A perturbation result of positivity is also presented. The biharmonic Green function is positive for the unit ball. This remains so under certain small perturbations of the ball. In dimension 2, perturbing the polyharmonic operator preserves positivity of the biharmonic Green function.

“Semilinear Problems” (Chapter 7) concerns the elliptic polyharmonic reaction-diffusion-type model equation mostly under Dirichlet boundary conditions not allowing decomposition in a system of second order Poisson equations with Dirichlet conditions. But also Navier and Steklov boundary conditions are looked at. The nonlinearity is assumed to be continuous and nondecreasing in order to prove existence and positivity results. Especially superlinear ones of the kind \[ f(u)=\lambda u+|u|^{p-1}u, \;\;\lambda\in \mathbb{R}, \;\;1<p, \] are investigated. Kernel estimates and monotonicity properties of the polyharmonic Green function are used to prove radial symmetry of positive solutions under Dirichlet conditions. Existence and nonexistence results for nontrivial solutions to the Dirichlet problem depend essentially on the value of the exponent \(p\). Assuming the space dimension exceeds the order of the equation, the sub-, super-, and critical case itself are studied. Much attention is paid to the geometrically relevant and complicated critical case. Here existence results are seen to depend strongly on the space dimension, the subcritical perturbations, and the geometry of the domain. For the supercritical equation only the bi-Laplace operator is considered, while the subcritical case turns out as not problematic. The critical growth equation is also studied under Navier and under Steklov boundary conditions. Here different phenomena occur.

Chapter 8, ‘’Willmore Surfaces of Revolution” presents an existence result for a priori bounded classical solutions to the Dirichlet problem for Willmore surfaces.

That the authors are experienced researchers on the topic of the volume becomes evident from the excellent, clear and well motivated presentation. Reading the closing sections “Biographical Notes” of chapters 2 to 8, together with the extensive list of 423 references, the engagement of each of the 3 authors becomes obvious. Whoever is interested in an exciting subject of the modern theory of higher order elliptic equations is recommended to study this exposition.

Reviewer: Heinrich Begehr (Berlin)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35J30 | Higher-order elliptic equations |

31B30 | Biharmonic and polyharmonic equations and functions in higher dimensions |

35J40 | Boundary value problems for higher-order elliptic equations |

35J61 | Semilinear elliptic equations |

35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |

74B05 | Classical linear elasticity |