This chapter provides the list of symbols, some elementary inequalities such as Young's inequality, Holder's inequality, and Jensen's inequality and domains with a conical point. The chapter also explains the quasi-distance function and its properties, Holder and Sobolev spaces, and function spaces. Function spaces are divided into two spaces: (1) Lebesgue spaces and (2) regularization and approximation by smooth functions.

This chapter explains the classical hardy inequalities, the Poincare inequality, the Wirtinger inequality, Hardy–Friedrichs-–Wirtinger type inequalities, and other auxiliary integral inequalities. The Wirtinger inequality has two conditions—namely, Dirichlet boundary condition and Robin boundary condition. Dirichlet boundary condition considers the problem of the eigenvalues for the Laplace–Beltrami operator on the unit sphere. Hardy–Friedrichs–Wirtinger type inequalities describe very general assumptions on the structure of the boundary of the domain in a neighborhood of the boundary point.

This chapter explains the Dini estimates of the generalized Newtonian potential, the equation with constant coefficients (Green's function), and the Laplace operator in weighted Sobolev spaces.

This chapter explains the Dirichlet problem in general domains, the Dirichlet problem in a conical domain, smoothness in a Dini–Liapunov region, and unique solvability results. The Dirichlet problem in general domains is explained with unique solvability, Alexandrov's maximum principle, the E. Hopf strong maximum principle, comparison principle, and Local maximum principle. The Dirichlet problem in a conical domain has different cases such as estimates in weighted Sobolev spaces and the power modulus of continuity. The Dirichlet problem is also explained with the help of some examples and higher regularity results. Smoothness in a Dini–Liapunov region studies the strong solutions in a Dini–Liapunov region. A unique solvability result investigates the existence of solutions in weighted Sobolev spaces for the boundary-value problem under minimal assumptions on the smoothness of the coefficients.

This chapter studies the behavior of weak solutions of the Dirichlet problem for a second-order elliptic equation in a neighborhood of a boundary point. The exponent is the best possible for domains with the assumed boundary structure in that neighborhood. The chapter describes a general assumption on the structure of the domain boundary in a neighborhood of the boundary point. The chapter also explains the estimate of the weighted Dirichlet integral, local bound of a weak solution, Holder continuity of weak solutions, and weak solutions of an elliptic inequality. The chapter introduces Dini continuity of the first derivatives of weak solutions, which is explained with the help of three factors: (1) local Dini continuity near a boundary smooth portion, (2) Dini-continuity near a conical point, and (3) global regularity and solvability.

This chapter studies the properties of strong solutions of the Dirichlet problem for nondivergent semilinear uniformly elliptic second-order equations in a neighborhood of a conical boundary point. The chapter illustrates the estimate of the solution modulus and recall the well-known comparison principle. The chapter also studies the properties of weak solutions of the Dirichlet problem for the divergence semilinear uniformly elliptic second-order equation in a neighborhood of conical boundary point.

This chapter investigates the behavior of solutions of the Dirichlet problem for a uniformly elliptic quasilinear equation of second order of nondivergence form near a corner point of the boundary of a bounded plane domain. The chapter assumes that the coefficients of the equation satisfy minimal conditions of smoothness and coordinated growth (no higher than quadratic) modulo the gradient of the unknown function. The estimate of the Nirenberg type is explained with formulation of the problem and the main result, the Nirenberg estimate, the behavior of the solution near a corner point, the weighted integral estimate, and proof of theorem. The chapter also introduces estimates near a conical point, which shows the behavior of strong solutions to the Dirichlet problem for uniform elliptic quasi-linear second order equation of non-divergent form near an angular point of the boundary of a plane bounded domain. Estimates near a conical point also reviews the barrier function, the weak smoothness of solutions, estimates in weighted spaces that establish the best-possible weighted exponent, and higher regularity results. The chapter closes with solvability results.

This chapter investigates the behavior of weak solutions to the Dirichlet problem for uniformly elliptic quasilinear equations of divergence form in a neighborhood of a boundary conical point. The chapter explains four principles: (1) the Dirichlet problem in general domains, (2) the m-Laplace operator with an absorption term, (3) estimates of weak solutions near a conical point, and (4) integral estimates of second weak derivatives of solutions, which is explained with the help of three factors—namely, local interior estimates, local estimates near a boundary smooth portion, and the local estimate near a conical point.

This chapter devotes the estimate of weak solutions to the boundary-value problems for elliptic quasi-linear degenerate second-order equations and investigates the behavior of weak solutions of the first- and mixed-boundary-value problems for quasi-linear elliptic equation of the second order with triple degeneracy and singularity in the coefficients in a neighborhood of singular-boundary point. The chapter illustrates the main theorem and proves a weak comparison principle for the quasi-linear equation, which extend the corresponding results. The chapter also introduces the strong maximum principle of E. Hopf, the boundedness of weak solutions, the construction of the barrier function, and the estimate of weak solutions in a neighborhood of a boundary edge, which uses the barrier function. The chapter concludes with a discussion on the Dirichlet problem, mixed problem, and proof of the main theorem.

This chapter investigates the behavior of strong solutions to the Robin boundary value problem for the second-order elliptic equations (linear and quasi-linear) in the neighborhood of a conical boundary point and obtains the best possible estimates of the strong solutions for the problems near a conical boundary point. The chapter explains the linear problem with the help of formulation of the main result, the comparison principle, the barrier function, global integral weighted estimate, local integral weighted estimates, and the power modulus of continuity at the conical point for strong solutions.

Let M be a differentiable manifold of dimension n of class C∞. Let p: V(M)→M be the fibre bundle of non-zero tangent vectors to M, with fibre type Rn-{0} with structure group GL(n, R) the general linear group in n real variables. We denote by π: W(M)→M the fibre bundle of oriented directions tangent to M. Let E(M) be the linear fibre bundle of frames on M and p−1E(M) the induced fibre bundle of E(M) by p. An infinitesimal connection on p−1E(M) is called a linear connection of vectors ([1]). The study of this connection leads us to single out a condition of regularity (§5). In this case, independent tensor forms can be introduced on V(M). To a regular linear connection of vectors are associated canonically two torsion tensors S and T as well as three curvature tensors R, P and Q; we find expressions for them in (§7). In view of obtaining the formulas of the habitual linear connections we establish a reduction theorem (§8). With the help of covariant derivations of two types ∇ and ∇' we form three Ricci identities for a vector field in the large sense (§9). In §10 we show that there exist betwen the two torsion tensors S and T as well as among the three curvature tensors R, P and Q of a general regular connection five identities called Bianchi identities. We then give explicit formulas for them.

A metric manifold is defined by the aata of a tensor field gij in the restricted sense of degree zero on W(M). To this tensor field is associated a scalar of degree two in the restricted sense F2=gij(x, v) vivj where F>0 is by definition the length of tangent vectors v to M at x∈M. With the help of g we can define the scalar product of two vectors of Tπy, y∈W(M), consequently an orthonormal frame on Tπy. Let us denote by E(W, g) the principal fibre bundle on W(M) of orthonormal frames. An infinitesimal connection on E(W, g) is called a Euclidean connection. We give the necessary and sufficient conditions in order that a linear connection on W(M) is naturally associated to a Euclidean connection of directions. We say that the datum of a function F>0 homogeneous of degree one on V(M) defines a Finslerian metric if it leads to a regular problem of the calculus of variations. The following result is the fundamental theorem of Finslerian Geometry: Given a Finslerian manifold there exists a regular Euclidean connection such that its torsion tensor S vanishes and the tensor T satisfies a condition of symmetry. Such a characterization of the Finslerian connection leads us naturally to Cartan's Euclidean connection ([1c], [13]). Using the results of chapter 1 we establish the fundamental formulas of Finslerian geometry the three curvature tensors, five Bianchi identities are completely made explicit[§ 8]. §9 is devoted to the semi-metric connection and we give a characterization of Berwald connection. We show that there exists an infinity of torsion-free connections of directions attached to F that define the same splitting of the tangent bundle as the Finslerian connection. These connections, differ from Berwald or Cartan connections by a homogeneous tensor tjk i of degree zero satisfying toj i=0-tko i, and have the same flag curvature as Berwald and Cartan connections.

This chapter is devoted to the study of infinitesimal isometries of a compact Finslerian manifold without boundary, and affine infinitesimal transformation of a regular linear connection of directions. ([1], [1a], [1b]). We recall the calculus rules of Lie derivatives of a tensor field in the large sense and of a form of a regular linear connection of vectors. Let L be the Lie algebra of infinitesimal transformations of M. To an X∈L is associated a certain endomorphism Ax of Tpz whose expression contains the torsion tensor T. Let Az (L) be the Lie algebra of endomorphisms of Tpz corresponding to the elements of L. We establish then a relation between A[X, Y]. AX, AY and the curvature of the linear connection, generalizing from the Riemannian case due to B. Kostant ([26], [1]). For the study of compact Finslerian manifolds we establish the divergence formulas for the horizotal 1-forms and for the vertical 1-forms on W(M) ([1],[2]). Next we study the 1-parameter group of infinitesimal transformations that leave invariant the splitting of the tangent bundle defined by Finslerian connection. We give a local characterization of isometries. In case M is compact and without boundary we prove the largest connected group of transformations that leave invariant the splitting defined by the Finslerian connection coincides with the largest group of isometries. We establish a formula linking the square of the vertical part of the lift of an isometry X on V(M) and the integral involving an expression of the flag curvature (R(X, uu,X). If this form is negative definite the isometry group is finite. Finally, to every infinitesimal isometry X of a Finslerian manifold is associated an antisymmetric endomorphism whose square of the module modulo a divergence puts in evidence a quadratic form ϕ depending on two Ricci tensor Rij and Pij.[1b]. We determine the conditions on them so that the isometry group of the manifold is finite. We study the particular case of Pij=0 In paragraph §10 we give a characterization of affine infinitesimal transformation (respectively partial) of regular linear connections of vectors. In paragraph § 11 we show that the Lie derivative L(X) commutes with the covariant derivatives of two types ▿ and ▿• (respectively of type ▿) when X defines an affine infinitesimal transformation (respectively partial), and conversely. Let L be the Lie algebra of affine transformations of a generalized linear connection, and L ˜ its lift on V(M). The Lie algebra Az(L), corresponding to L is the Lie algebra of a connected group Kz(L) of linear transformations of Tpz. The study of this group is the objective of paragraphs § 12, 13 and 14. In the case when the Lie algebra L ˜ is transitive on V(M) we have a relation of inclusion σz⊂Kz(L)⊂No(σz) where σz is the group of restricted homogeneous holonomy at z∈V(M) and No(σz) indicates the passage to the connected normalizer [1]. The group Kz(L) has been introduced by B. Kostant[26] in the Riemannian case. In conclusion we also study the case of affine infinitesimal transformation of a Finslerian connection.

Let (M, g) be a compact Finslerian manifold, of dimension n, and let W(M) be the fibre bundle of unitary tangent vectors to M. Let φ(x, v) be a function on W (M). We now introduce the Laplacian Δφ=Δφ(x, v) of φ(x, v). Now Δφ decomposes into two parts, Δ ϕ = Δ ¯ ϕ + Δ · ϕ. . We call Δ ¯ ϕ the horizontal Laplacian, and Δ · ϕ the vertical Laplacian. To the function φ(x, v) on W(M) we associate a symmetric 2-tensor Aij(φ), and we establish a formula linking the square of Aij(φ) to Δ ¯ ϕ and aquadratic form Ф(ϕ*, ϕ· ), where ϕ* is the horizontal derivative and ϕ· the vertical derivative of φ, with coefficients that depent on the curvature of the Finslerian connection. Imposing a certain condition on the curvature tensor, and in case the vertical Laplacian vanishes, we obtain an estimate for the function λ of Δ ¯ ϕ ( Δ ¯ ϕ = λ ϕ ) . More precisely, λ cannot always be between zero and n.k where k is a constant>0. In case M is simply connected and λ=n.k, then (M, g) is homeomorphic to an n-sphere. This is a generalization of a theorem of Lichnerowicz-Obata in the Riemannian case. The rest of the work is devoted to the deformation of a Finslerian metric Let Fo(M, gt) be a deformation of (M, g) preserving the volume of W(M). We prove that the critical points g0∈Fo(gt), of the integral I(gt) on W(M) of a certain Finslerian scalar curvature define a Generalized Einstein manifold [6]. We evaluate the second variation of I(gt) at the critical point g0 and show that in certain cases and for an infinitesimal conformal deformation, we have I″(g0)>-0.

The objective of this chapter is to obtain a classification of Finslerian maniforlds. Let (M, g) be a Finslerian manifold of dimension n, and W(M) the fibre bundle of unit tangent vectors to M. The curvature form of the Finslerian connection (Cartan) associated to (M, g) is a two from on W(M) with values in the space of skew-symmetric endomorphisms of the tangent space to M. It is the sum of three two forms of type (2,0), (1,1) and (0,2) whose coefficients R, P and Q constitute the three curvature tensors of the given connection. In the first part we study the Landsberg manifolds, manifolds with minimal fibration and Berwald manifolds. The manifold M is called a Landsberg manifold if P vanishes everywhere. This condition is equivalent to the vanishing of the covariant derivative in the direction of the canonical section v: MV(M) of the torsion tensor. For a Riemannian metric (0,2) on V(M) this condition means that for every xεM the fibre p−1(x) becomes a totally geodesis manifold where p: V(M) M (see [5 and §7]). We examine the case when V(M)M is of minimal fibration as well as when M is a Berwald manifold. When M is compact and without boundary we put some global conditions on the first curvature tensor R or flag curvature of the Cartan connection. In the second part we study by deformations the metric of compact Finslerian manifolds in order that their indicatrix become Einstein manifolds.

This chapter is a study of isotropic and constant sectional curvature Finslerian manifolds. We first recall briefly the basics of Finslerian manifolds, define the isotropic manifolds and single out the properties of their curvature tensors. We then give a characterization of Finslerian manifolds with constant sectional curvature, generalizing Schur's classical theorem. We next determine the necessary and sufficient conditions for an isotropic Finslerian manifold to be of constant sectional curvature. Our conditions bear on the Ricci directional curvature or on the second scalar curvature of Berwald. We show that the existence of normal geodesic coordinates of class C2 on isotropic manifolds forces them to be Riemannian or locally Minkowskian. We also deal with the case of compact isotropic Finslerian manifolds with strictly negative curvatures. In chapter III we give a classification of complete Finslerian manifolds with constant sectional curvatures. We prove that all geodesically complete Finslerian manifolds of dimension n>2 with negative constant sectional curvature (K2 with strictly positive constant sectional curvature and whose indicatrix is symmetric and has a scalar curvature independent of the direction is homeomorphic to an n-sphere In the case when the Berwald curvature H vanishes and torsion tensor as well as its covariant vertical derivative are bounded we prove that the manifold in question is Minkowskian. In the last chapter we establish the ‘axioms of the plane’. By defining the totally geodesic, semi-parallel and auto-parallel Finslerian submanifolds we establish the criteria that permit to identify if a Finslerian manifold is of constant sectional curvature in the Berwald connection (axiom 1), in the Finslerian connection (axiom 2) or is Riemannian (axiom 3).

We give a characterization of the projective vector fields on Finslerian unitary tangent fibre bundle and we prove that in the compact case the existence of projective vectors is related to the sign of the flag curvature. We define the notion of a restricted infinitesimal projective transformation, and introduce a new projective invariant tensor. We determine the necessary and sufficient conditions for the vanishing of projective invariants. We study the case when the Ricci directional curvature (R.D.C.) is satisfied under certain conditions especially when the curvature is constant. (This is a generalization of Einstein manifold). We show that any simply connected, metrically complete. Finslerian manifold with a R.D.C. positive constant and admitting a proper vector field leaving the covector of torsion trace invariant is homeomorphic to an n-sphere.

Let X be a vector field over M and exp(tX)the local 1-parameter group generated by X and exp(tX^ ) its lift to V(M). We call X a conformal vector field or a conformal infinitesimal transformation if there exists a function φ on M such that the Lie derivative of the metric tensor g is equal to 2 φ.g. To every field of co-vectors Y(z)εA1(W) is associated a covariant symmetric 2-tensor τ(Y). We calculate its square by making intervene explicitly the curvature of the space. This formula helps us characterize the infinitesimal conformal transformations when M is compact without boundary. In paragraph 4 we establish a formula giving the square of the vertical part of the lift of conformal vector field on V(M) in terms of the flag curvature and we show that there exists a non-trivial conformal vector field only if the integral of the quadratic form (R(X, u)u, X) is positive. Next we take up the case when the scalar curvature H ˜ = g j k H ˜ j k w h e r e 2 H ˜ j k = ∂ 2 H ( v , v ) / ∂ v j ∂ v k est a non-positive constant (H∼ =constant≤0.) and the torsion trace co-vector satisfies a certain condition. On making a hypothesis on the vertical Ricci curvature Qij we show that for dim M>2, and M compact, the largest connected group C0(M) of infinitesimal conformal transformations coincides with the largest connected group of isometry I0(M), thus generalizing the Riemannian case. We deal with case where the 1-form X=Xi(z)dx1 corresponding to the vector field is semiclosed and obtain in the conformal case the corresponding equations. Lett σ be a semi-closed 1-form whose co-differential δσ is independent of the direction and let Δ ¯ σ be the horizontal Laplacian and let λ(y), yεW(M) a the function such that Δ ¯ σ = λ ( y ) σ. . We give an estimate of λ(y) as a function of proper value of the flag curvature. The rest of the chapter is devoted to the lift to W(M) of the conformal vector field when its dual 1-form is semi-closed.