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Many early variational problems like Dido’s problem or the problem of the hanging chain have constraints in a natural way: Maximize the area with given perimeter or minimize the potential energy of a hanging chain with given length. These constraints belong to the class of isoperimetric constraints, and they are of the same type as the functional to be maximized or minimized.

In order to give the functionals $$ J(y) = \int ^b_a F(x,y, y') dx $$ a domain of definition, we need to introduce suitable function spaces. First of all we require that the Lagrange function or Lagrangian, $$ F : [a, b] \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}, {\qquad }\text {is continuous.} $$ Here \([a, b] = \{x|a\le x\le b\}\) is a compact interval in the real line.

The Euler-Lagrange calculus was created to determine extremals of functionals. If the solution of the Euler-Lagrange equation is unique among all admitted functions, then physical or geometric insights into the problem might lead to the conclusion that it is indeed the desired extremal. In addition, the second variation provides necessary and also sufficient conditions on extremals.

We consider the nonlinear eigenvalue problem posed by a parameter-dependent semilinear second-order elliptic equation on a bounded domain with the Dirichlet boundary condition. The coefficients of the elliptic operator are bounded measurable functions and the boundary of the domain is only required to be regular in the sense of Wiener. The main results establish the existence of an unbounded branch of positive weak solutions.

We study global symmetry-breaking equilibria of the van der Waals–Cahn–Hilliard phase-field model on the sphere \(S^{2}\subset \mathbb{R }^{3}\). Our primary motivation comes from observed two-phase phenomena for lipid-bilayer vesicles. Although we ignore the elastic deformability of the latter, this work represents an important first step in understanding global symmetry-breaking in such models. We combine symmetry methods with global bifurcation analysis for basic existence results, and then establish a-priori bounds and in some cases, nodal properties, to glean detailed information about global solution branches.

We investigate critical points of the functionalover a ball in ℝn. Here, W is radially symmetric but not convex. We embed the functional into a family of functionalswhere E0,0(u) = E(u). A global bifurcation analysis yields a branch of non-trivial critical points depending on λ and positive ε, where we can set λ = 0. The geometric properties preserved on that branch, due to the maximum principle, prove compactness such that the singular limit as ε ↘ 0 exists. Under natural conditions on W and G the critical point obtained in this way is a minimizer of the original functional. That plan can be carried out only under the restriction of radial symmetry, since the maximum principle applies only to special elliptic equations of fourth order. That restriction, however, is not essential since every minimizer of the functional is radially symmetric.

We present a theorem ensuring the existence of local solution branches for one-parameter bifurcation problems in which the linearization at the trivial solution possesses a two-dimensional kernel. In particular, we provide a straightforward “test” that is sufficient for the existence of local solution continua. We demonstrate our abstract theorem with several concrete examples for second-order systems of elliptic partial differential equations with symmetry.

We prove mathematical approximation results for the (hyperviscous) Burgers equation driven by additive Gaussian noise. In particular we show that solutions of ``approximating equations'' driven by a discretized noise converge towards the solution of the original equation when the discretization parameter gets small. The convergence takes place in the expected value of arbitrary powers of certain norms; i.e., all moments of the difference of the solutions tend to zero in certain function spaces. For the hyperviscous Burgers equation, these results are applied to justify the approximation of certain correlation functions that play a major role in statistical turbulence theory.

We establish the existence of time-periodic solutions of semi-linear wave equations on the unit sphere in R 3. The problem has been studied previously in (1982, V. Benci and Fortunato, Ann. Mat. Pura Appl.132, 215–242) using variational techniques. Our results here are much sharper: We employ delicate methods of bifurcation theory (1979, H. Kielhofer, J. Math. Anal. Appl.68, 408–420; 1987, H. Kielhofer and P. Kotzner, J. Appl. Math. Phys.38, 201–212) combined with well-known group-theoretic ideas to find branches of small-amplitude solutions. Precise spatio-temporal patterns of solutions are uncovered as a by-product of the analysis. Moreover, in certain cases we prove the existence of global solution branches (in the sense of P. Rabinowitz).

We investigate critical points of the free energy of the Cahn-Hilliard model of a binary alloy under the constraint of a constant mass. The domain is the unit square. Minimizers of the energy without interfacial energy term are given by a decomposition of the two components of the alloy, but the interfaces between the components are arbitrary. Specic patterns are only formed if an interfacial energy term is present. We select such patterns of minimizers by an approximation of se- quences of conditionally critical points of the free energy when the interfacial energy term tends to zero. This is what we call Pattern Formation of the Stationary Cahn- Hilliard Model. Mathematically it is a singular limit process. We obtain the conditionally critical points by a global bifurcation analysis of the Euler-Lagrange equation for the free energy where the mass is the bifurcation pa- rameter and where the constant homogeneous mixtures give the trivial solutions. By using characteristic symmetries and monotonicities of the bifurcating solutions we show that singular limits exist for all masses in the so-called spinodal region and that they are minimizers of the free energy without interfacial energy term. A binary alloy in a vessel can be described by a function u: ! IR as follows: u(x)2 (0; 1) means that the mixture contains u(x) 100 % of one component at x2 . The energy density of the alloy is modelled by W (u), where W is a two-well potential (Figure 1).

has been studied in many publications (the first global description was given in [19]). Nonetheless there were open questions some of which are answered in this paper. We motivate some of these questions now: For a suitable class of (smooth) fimctions f : IR+ ~ IR problem (1.1) has infinitely many solutions for 2 = 1. This was shown in [15] via the method of suband supersolutions which also gives an order of these solutions: 0 0 (which is not excluded in [15]) there is also a global branch ( = continuum) C + of solutions (2 ,u) bifurcating from the trivial solution line {(2, 0)} at 2 = 21 where #1 = 21 f ' ( 0 ) is the principal eigenvalue of the linearization

Der Euler-Lagrange-Kalkul wurde erfunden, um Extremale zu berechnen. Ist die Losung der Euler-Lagrange-Gleichung unter den zulassigen Funktionen eindeutig, kann mit physikalischen oder geometrischen Einsichten in das Problem geschlossen werden, dass sie die gesuchte Extremale ist. Daruber hinaus konnen mithilfe der zweiten Variation weitere notwendige und auch hinreichende Bedingungen gepruft werden.

Schon bei den klassischen Problemen wie dem der Dido oder der hangenden Kette treten in naturlicher Weise Nebenbedingungen auf: Maximiere die Flache bei gegebenem Umfang, minimiere die potentielle Energie der hangenden Kette bei gegebener Lange. Diese Nebenbedingungen gehoren zu der Klasse der isoperimetrischen Nebenbedingungen und sind vom gleichen Typ wie das zu maximierende oder zu minimierende Funktional.

We consider a bifurcation problem for a general class of fully nonlinear, second-order elliptic equations on a regular bounded domain in R n {\mathbb {R}^n} and subject to homogeneous Dirichlet boundary data. We assume that the linearized problem about the trivial solution possesses a positive solution for at least one isolated parameter value. With no other growth or sign conditions imposed upon the nonlinearity, we establish the existence of a global branch of nontrivial positive solutions. Moreover, if there is only one such isolated value of the parameter, we deduce that the branch of positive solutions is unbounded.

We study the Cahn-Hilliard energy E ɛ(u) over the unit square under the constraint of a constant mass m with (ɛ > 0) and without ɛ= 0) interfacial energy. Minimizers of E 0(u) have no preferred pattern and we select patterns via sequences of conditionally critical points of E ɛ(u) converging to minimizers as ɛ tends to zero. Those critical points are not minimizers if the singular limit has no minimal interface. We obtain them by a global bifurcation analysis of the Euler-Lagrange equations for E ɛ(u) where the mass m is the bifurcation parameter. We make use of the symmetry of the unit square, and the elliptic maximum principle, in turn, implies that the location of maxima and minima is fixed for all solutions on global branches. This property is used to guarantee the existence of a singular limit and to verify the Weierstrass-Erdmann corner condition which proves its minimizing property.

In this paper we prove the existence of doubly periodic solutions of certain nonlinear elliptic problems on R 2 and study the geometry of their nodal domains. In particular, we will show that if we perturb a nonlinear elliptic equation exhibiting a small amplitude doubly periodic solution whose nodal domains form a checkerboard pattern, then the perturbed equation will have a unique nearby solution which is still doubly periodic, but for which the nodal line structure breaks up. Moreover, we indicate what can happen if we start with a large amplitude doubly periodic solution whose nodal domains form a checkerboard pattern, and we relate these solutions to the Cahn-Hilliard equation and spinodal decomposition.

We prove a multiplicity result for positive solutions of a class of semilinear elliptic Dirichlet problems Lu + f(u) = 0 over bounded domains via sub- and supersolutions. A concrete example for the strictly increasing nonlinearity f is given, too.

The smallest eigenvalue of Δv+λf′(0)v=0 gives rise to a global positive branch of Δu+λf(u)=0 in Ω⊂ℝ2 together with homogeneous Dirichlet boundary conditions (f(0)=0,f′(0)≠0). The theory, however, guarantees only that this branch is a continuum. We present a result that in a reasonably large class of problems this branch is actually an unbounded smooth curve. In the second part we study the asymptotic behavior of this positive smooth curve whenf has exponential growth. All results apply also to branches emanating at higher eigenvalues. As shown in [7] they have a fixed nodal structure when restricted to appropriate fixed-point spaces.

In this paper we study a general class of quasi-linear elliptic equations of the form $$ a_{ij} (\nabla u,u)u_{x_i x_j } + g(\lambda \nabla u,u) = 0{\text{ }}in{\text{ }}\Omega $$ (1.1) , where Ω ⊂ ℝ2 is a bounded domain, u: Ω → ℝ, “∇(†)” denotes the gradient, and λ e ℝ. On the boundary aΩ we impose fully nonlinear Neumann conditions: $$ q(\nabla u,u) \cdot n|_{\partial \Omega } = 0 $$ (1.2) , where q(∇u,u) e ℝ2, n denotes the outward unit normal to ∂Ω, and “•” is the usual inner (dot) product on R2. Such problems arise naturally in continuum physics; (1.1) models a general class of nonlinear diffusion processes, and (1.2) is a zero-flux condition. In a recent work [5] we studied global bifurcation problems for (1.1) defined on all of Rn in the presence of lattice symmetry. In particular, (given certain hypotheses, which we recapitulate in Section 2) we showed that the precise nodal configuration of the eigenfunction (of the linearized problem at the trivial solution) is preserved along the associated global bifurcating solution branch.

Let X and Y be Banach spaces, Y ⊂X, and let V be a neighborhood of zero in Y. We consider the equation G(λ, u) ≡ A(λ)u + F(λ, u) = 0, where G: [−d1, d1] × V → X, G(λ, 0) = 0, and A(λ) is the Frechet derivative of G with respect to u at (λ, 0). Furthermore, we assume that G is analytic with respect to λ and u. Bifurcation at a simple eigenvalue means that zero is a simple eigenvalue of A (0). Let μ(λ) be the simple eigenvalue of the perturbed operator A(λ) for λ near zero. Let d j μ(0) dλ j = 0, j = 0,…, m − 1, d m μ(0) dλ m A m ≠ 0, or μ(λ) ≡ 0 . Under the nondegeneracy condition m = 1 the existence of a unique curve of solutions intersecting the trivial solution (λ, 0) at (0, 0) is well known. Furthermore the “Principle of Exchange of Stability” was established in this case. We show that in the degenerate case (m > 1) up to m bifurcating curves of solutions can exist and that at least one nontrivial curve exists if m is odd. Our approach supplies all curves of solutions near (0, 0) together with their direction of bifurcation and their linearized stability. The decisive fact is that Am is also the leading term of the bifurcation equation. A consequence is a “Generalized Principle of Exchange of Stability”, which means that adjacent solutions for the same λ have opposite stability properties in a weakened sense. For practical use we give a criterion for asymptotic stability or instability which follows from the construction of the curves of solutions themselves.

The “Principle of Reduced Stability” says that the stability of bifurcating stationary or periodic solutions is given by the finite dimensional bifurcation equation obtained by the method of Lyapunov-Schmidt. To be more precise, the linearized stability is governed by the linearization of the bifurcation equation about the bifurcating branch of solutions and in particular by the signs of the real parts of the perturbation of the eigenvalues along this branch. This principle is true for simple eigenvalue bifurcation whereas it may be false for higher dimensional bifurcation equations. A condition for the validity of that principle is given. A counterexample shows that it cannot be dropped in general.

On fait le point sur les resultats de bifurcation locale et globale et on considere la stabilite des branches de bifurcation

A semilinear parabolic initial-boundary-value problem of order 2m in a possibly unbounded domain Ωx(O,T), Ω⊂Rn, is considered within the framework of the Lp-and Cα-theory. In the first case a proof is given of the existence of a “strict” solution of the corresponding evolution equation. In the second case one can guarantee a classical solution, provided the homogeneous linear parabolic equation has a unique classical solution. Only local solvability is considered. The nonlinearity is a Holder-continuous function of the derivatives up to the order 2m-1 of the unknown solution. The principal tool is the semigroup-theory in Lp(Ω) as well as in Cα(\(\bar \Omega \)). In the latter case the semigroup is not strongly continuous, but it has sufficiently good properties to use it for existence proofs of classical solutions.