Chapter 6 is devoted to static bifurcation theory for degenerate Robin problems for semilinear second-order elliptic differential operators under two conditions on the boundary functions \(a(x^{\prime })\) and \(b(x^{\prime })\) : Our approach here is based on the Semenov approximation. This chapter is the heart of the subject. In Sect. 6.1 we reformulate Theorem 1.4 as Remark 6.2 and Theorem 6.3 from the viewpoint of linear partial differential equations. The proof of Theorem 6.3 will be given in detail in the next Chap. 7 , due to its length. In Sect.6.2, as an application of Theorem 6.3 we study local static bifurcation problems for a class of semilinear Robin problems, and state a local bifurcation Theorem (Theorem 6.4). The proof of Theorem 6.4 will be given in detail in Chap. 8 , due to its length. Section 6.3 is devoted to global static bifurcation theory for semilinear Robin problems. In this section, we consider the following semilinear Robin problem: For a given function \(f(x,\xi )\) defined on \(\overline {D} \times [0, \infty )\) , find a non-negative function \(u(x)\) in D such that 5 \(\displaystyle \begin {cases} Au = f(x,u) &\mbox{in }D, \\ B{\boldsymbol \gamma }u = a(x^{\prime })\frac {\partial u}{\partial {\boldsymbol \nu }} + b(x^{\prime })u = 0 &\mbox{on }\partial D. \end {cases} \) The semilinear Robin problem (6.5) is the prototype of a class of semilinear second order elliptic boundary value problems which arise in numerous applications in physical problems and in problems of Riemannian geometry. A general existence principle in many cases important in practice is the method of super- and subsolutions or the monotone iterative method (Theorem 6.7). Namely, the existence of both a supersolution and a subsolution yields the existence of a solution with effective error methods (the principle of super- and subsolutions). Moreover, if we know several super- and subsolutions, then we can obtain at least three solutions (the multiplicity principle). We remark that Theorem 6.7 is a generalization of Amann (1976) and de Figueiredo (1982) to the degenerate Robin case under the conditions (H.1) and (H.2). In order to formulate our uniqueness theorem of positive solutions of the semilinear Robin problem (6.5), we introduce another fundamental condition, sublinearity, on the nonlinear term \(f(x,\xi )\) : (SL) For each \(0 < \tau < 1\) , we have the inequalities S1 \(\displaystyle f\left (x,\tau \xi \right ) \geq \tau \,f(x,\xi ) \quad \mbox{for all }x \in \overline {D}\mbox{ and }\xi > 0, \) and S2 \(\displaystyle f(x,0) \geq 0 \quad \mbox{for all }x \in \overline {D}. \) Then our uniqueness theorem for the semilinear Robin problem (6.5) is stated as Theorem 6.8. Our approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of pseudo-differential operators, which may be considered as a modern version of the classical potential approach. The theory of pseudo-differential operators is one of the most influential works in the modern history of analysis, and is a very refined mathematical tool whose full power is yet to be exploited.

Semilinear Hypoelliptic Robin Problems via Semenov Approximation

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发表日期：

2025年

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Kazuaki Taira

摘要

Chapter 6 is devoted to static bifurcation theory for degenerate Robin problems for semilinear second-order elliptic differential operators under two conditions on the boundary functions \(a(x^{\prime })\) and \(b(x^{\prime })\) : Our approach here is based on the Semenov approximation. This chapter is the heart of the subject. In Sect. 6.1 we reformulate Theorem 1.4 as Remark 6.2 and Theorem 6.3 from the viewpoint of linear partial differential equations. The proof of Theorem 6.3 will be given in detail in the next Chap. 7 , due to its length. In Sect.6.2, as an application of Theorem 6.3 we study local static bifurcation problems for a class of semilinear Robin problems, and state a local bifurcation Theorem (Theorem 6.4). The proof of Theorem 6.4 will be given in detail in Chap. 8 , due to its length. Section 6.3 is devoted to global static bifurcation theory for semilinear Robin problems. In this section, we consider the following semilinear Robin problem: For a given function \(f(x,\xi )\) defined on \(\overline {D} \times [0, \infty )\) , find a non-negative function \(u(x)\) in D such that 5 \(\displaystyle \begin {cases} Au = f(x,u) &\mbox{in }D, \\ B{\boldsymbol \gamma }u = a(x^{\prime })\frac {\partial u}{\partial {\boldsymbol \nu }} + b(x^{\prime })u = 0 &\mbox{on }\partial D. \end {cases} \) The semilinear Robin problem (6.5) is the prototype of a class of semilinear second order elliptic boundary value problems which arise in numerous applications in physical problems and in problems of Riemannian geometry. A general existence principle in many cases important in practice is the method of super- and subsolutions or the monotone iterative method (Theorem 6.7). Namely, the existence of both a supersolution and a subsolution yields the existence of a solution with effective error methods (the principle of super- and subsolutions). Moreover, if we know several super- and subsolutions, then we can obtain at least three solutions (the multiplicity principle). We remark that Theorem 6.7 is a generalization of Amann (1976) and de Figueiredo (1982) to the degenerate Robin case under the conditions (H.1) and (H.2). In order to formulate our uniqueness theorem of positive solutions of the semilinear Robin problem (6.5), we introduce another fundamental condition, sublinearity, on the nonlinear term \(f(x,\xi )\) : (SL) For each \(0 < \tau < 1\) , we have the inequalities S1 \(\displaystyle f\left (x,\tau \xi \right ) \geq \tau \,f(x,\xi ) \quad \mbox{for all }x \in \overline {D}\mbox{ and }\xi > 0, \) and S2 \(\displaystyle f(x,0) \geq 0 \quad \mbox{for all }x \in \overline {D}. \) Then our uniqueness theorem for the semilinear Robin problem (6.5) is stated as Theorem 6.8. Our approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of pseudo-differential operators, which may be considered as a modern version of the classical potential approach. The theory of pseudo-differential operators is one of the most influential works in the modern history of analysis, and is a very refined mathematical tool whose full power is yet to be exploited.