Partial Differential Equation Formulations from Variational Problems

The calculus of variations applied in multivariate problems can give rise to several classical Partial Differential Equations (PDE’s) of interest. To this end, it is acknowledged that a vast range of classical PDE’s were formulated initially from variational problems. In this paper, we aim to formulate such equations arising from the viewpoint of optimization of energy functionals on smooth Riemannian manifolds. These energy functionals are given as sufficiently regular integrals of other functionals defined on the manifolds. Relevant Banach domains which contain the optimal functional solutions are identified by preliminary analysis, and then necessary optimality conditions are discovered by differentiation in these Banach spaces. To determine specific optimal functionals in simple settings, smaller target domains are taken as appropriate subsets of the Banach (Sobolev) spaces. Briefings on analytical implications and approaches proffered are included for the aforementioned simple settings as well as more general case scenarios.


Introduction
A wide class of Partial Differential Equations are formulated initially from problems in the calculus of variations. In the bid to estimate critical points of functionals on subsets of infinite dimensional linear spaces, PDE's are often generated by formulation laws of the calculus of variations. Local and global minima of such functionals have profound implications in models of the physical universe. For instance, critical points of a functional usually correspond to the equilibrium stages of associated physical systems, and a non-exaggerated perturbation of a system from a state of equilibrium tends to return it to the same equilibrium state. Two energy functionals (Dirichlet and Perelman) are probed for their critical points in simple settings. The local minimum of Perelman's Energy as included could have new contributions towards the collection of static Ricci solitons currently known. Hence, this paper serves as an appendage to the very modern study of Ricci Flow. The section on Dirichlet's Energy probes into the very foundations of Sobolev and Lebesgue spaces.
In due course, we will invoke two fundamental tools during formulation which are outrightly stated in this section. The first is a general theorem in functional optimization theory and the second is a lemma of variational calculus.
Optimization Theorem [2] -Let E be a real reflexive Banach space, and the functional f: E ᴜ {+∞ } be convex, lower semi-continuous and proper. Then, i.) For any non-empty subset K of E that is weakly compact (closed, convex and norm-bounded), there exists α K such that min ; ii.) If in addition f is coercive, there exists such that min . [6] -Let Ω be a regular, orientable and bounded submanifold-with-boundary. Let Ω and assume that

Lemma of The Calculus of Variations
. Ω 0 for all Ω . Then 0 on Ω , where Ω is of dimension m and is the geometric m-volume on Ω. It is rather straightforward to prove the above stated lemma by contradiction, as hence shown. Assume | | " 0 for some Ω\$Ω, where $Ω is the boundary of the closure of Ω defined in the geometric sense. Then for some % > 0, we have ( , $Ω) < δ and |ℎ( )| ≥ |*( + )| , ∀ ∈ .( , δ) ∩ Ω. Setting: where { , , , … , H } is an orthonormal basis of functions for the tangent spaces to Ω at each point and [ , , , … , H ] N is the usual orientation for O N Ω at each P ∈ Ω. We can draw the same conclusions by adjusting certain conditions of the lemma, such as taking Ω to be a submanifold-without-boundary and using ∈ (Ω Q ) as the test functions, when appropriate.
The function spaces (Ω Q ) and (Ω) are not reflexive, which will prompt us to consider instead weak formulations of the optimization problems during computations. That is to say, we will target solutions in larger reflexive Sobolev spaces, reckoning with the fact that (Ω Q ) is dense in R S,N (Ω) for any natural k and for 1 < P < ∞. Occasionally, weak solutions also turn out to be solutions of the classical problems. Following our presentation of the fundamental tools, we now give two methods of formulation of classical PDE's associated to optimization of differentiable functionals.

Method 1 [6]
Let Ω ⊆ ℝ be a connected, orientable and bounded submanifold of class , , and we are to minimize the functional We will choose W such that it accommodates appropriate tangent cones, which are variations of the form Y + d for Y ∈ W and ∈ (Ω) (resp (Ω Q )). There exists a positive The regularity of [ at Y_ is necessary for passing the limit into the above integral as we have done in the second line. This gives us uniform convergence of the integrand by way of the mean value inequality, since the integrand equals The reader ought to reckon with the divergence operator (div) invoked above, which will be used explicitly in the included illustrations. As such, we assume in addition that [ is of class , and we get that ∀ ∈ (Ω) (or (Ω Q ) if Ω Q is without boundary), at any local or global minimizer Y_ of on W, which gives us a necessary optimality condition. If Ω is an open subset of ℝ , then the above is simply written as

Method 2
Now, assume that Ω ⊆ ℝ is open, bounded with a regular topological boundary, and we are to minimize the functional yΩ Ω For this case, we work with test functions ∈ (Ω Q ) because of the contributor to the functional from the boundary, and our formulation yields the necessary optimality condition l[ m ( , Y_ , zY_ ).
Of course, we need the functions [ and w to be sufficiently regular. Applying Green's theorem to a term in the above formulation; at any local or global minimizer Y_ of . By the fundamental lemma of the calculus of variations, we conclude - which is a boundary value PDE problem of Neumann type. By slightly adjusting prior hypotheses, it is straightforward to generalize that the formulations from (1) and (2) give not only necessary optimality conditions for minimizers, but for local critical points at large. The given lemma of the calculus of variations applies to larger reflexive Sobolev spaces, and this we can infer from a generalization of the lemma called the du Bois -Reymond lemma. It gives us that for any ℎ ∈ R S,N (Ω) satisfying t ℎ( ). ( ) Ω = 0 for all ∈ (Ω) then ℎ ≡ 0 almost everywhere on Ω. However, it is most convenient to work with subsets of the reflexive Hilbert space R S,, (Ω) because of continuity of the partial inner product -Eℎ, F = ℎ( ). ( ) Ω …b ℎ, ∈ R S,, (Ω) and the availability of other analytical solution tools such as the Lax-Milgram theorem [3]. Here, given ℎ ∈ R S,, (Ω) satisfying t ℎ( ). ( ) Ω = 0 for all ∈ R S,, (Ω) , then ℎ ≡ 0 almost everywhere on Ω. We will consider the weakly formulated methods in these larger reflexive Sobolev spaces in order to investigate existence and/or uniqueness of solutions to the optimization problems. Given the possibility of solution existence from the optimization theorem, we proceed to solve the weakly formulated PDE in a Sobolev space, and finally check whether the weak solutions are also solutions in (Ω Q ). It may turn out that the weakly formulated problem has a solution, while the classical problem does not. In the succeeding examples, we will illustrate the theoretical framework laid out above.

Result 1: Perelman Energy Functional
Let x ⊂ ℝ be a regular, compact and connected hypersurface. Perelman's energy functional on x is formally analogous to heat flow along the manifold and we give it by where † is the scalar curvature of x. In any setting, the functional lacks coercivity, and usually it also lacks a global minimizer.
As an illustration of this statement, let x be a regular surface in ℝ ‹ consisting only of elliptic points, in which case the scalar curvature equals twice the Gaussian curvature and the Perelman energy is strictly positive for any Y. We consider the problem in weak settings in order to investigate arguments using our optimization theorem. R can be any of the classical Sobolev spaces containing (x), and we see that inf m∈• (Y) = 0 by taking ||Y|| • to infinity along the positive direction of constant functionals Y ≡ Ž , where Nevertheless, the formulation of method 1 in subsection 2.1 above provides weak local critical points of which we hereby discuss. This formulation gives us at any critical point Y_ of . Observe that (3) is set as a non-linear second order P. D. E, as the divergence operation on the right hand side produces a second order differential of Y_ . As a simple computational illustration, we will have x to be the two-dimensional unit sphere x , ⊂ ℝ ‹ . This is a compact surface consisting only of elliptic points embedded in the real Euclidean 3-space with unit Gaussian curvature • ≡ 1 so that its scalar curvature is also constant: † ≡ 2.
We have , , to be the unit vectors in the directions of the partial derivatives Φ oe ' and Φ oe ' respectively. Observe that we have used the elementary property of directional derivatives; ∇ i-® = ∇ -® for a scalar field . Hence, the formulation in (3)  which is a point. However, continuous differentiability on the entire compact unit sphere is not obtainable.
The nature of critical functions can be investigated using the second variation test [7] of the functional . We can judge the nature of Y_ by considering variations of the form Y_ + ¼ for ¼ > 0 small enough to judge how acts locally around Y_ . We may examine the most significant terms of an associated Taylor Series expansion for this purpose. The smoothness of the metric of the hypersurface x guarantees that is more than twice differentiable, so that we can deduce the required terms as follows.
In such an event, the critical function Y_ would be a strict local minimizer for the Perelman energy. Using the weak solution discussed above, (wherein the spherical coordinate system was employed and oe ' set to zero) this situation can be created simply by truncating parallels ¡ , * of x , for which |´(¡ , * )| > √2, for 5 (¡ , ) =´ as previously defined. The resulting manifold would be a connected spherical section with boundary, and the critical function would then be a classical solution to (3) above.
In the classical applications of Perelman's energy, the functional Y is time dependent and critical points of with respect to time characterize steady Ricci Solitons. Therefore, Perelman's entropy can be regarded as a variational tool for the study of Ricci flow, which is an advanced current area of interest in pseudo-Riemannian geometry. It is also worthy of note that the same functional also occurs quite importantly in a branch of theoretical physics known as string theory [8].

Result 2: Dirichlet Energy Functional
Let Ω ⊂ ℝ be open, bounded and with topological boundary. The Dirichlet energy functional on Ω is given by and ℎ is a particular differentiable function defined on the compact set $Ω. The classical problem is to minimize over W , but we first consider the weak setting in the reflexive Sobolev space R ,, (Ω) ≔ AE (Ω) to effectively perform analysis using the given optimization theorem in the Introduction section. In particular, this setting is made appropriate by continuous differentiability of the functional on AE (Ω). Hence, the domain W′ of in this setting is the pre-image of the singleton ℎ ∈¸,(∂Ω) under the continuous trace operator; e : AE (Ω) ⟶¸,(∂Ω) so W′ is (norm-) closed in AE (Ω). Moreover, the set W′ is convex because For any function Y in W 5 , it is easy to check that the set . = 9Y ∈ W 5 : (Y) ≤ (Y ); is bounded due to the coercivity of , giving us existence of a minimizer for on . and thus also on W′. The critical function Y_ will exist uniquely due to strict convexity of .
Given the minimizer Y_ ∈ AE (Ω), the weak formulation for this problem is the following boundary value PDE: This is obtained by implementing the formulation method 2 (subsection 2.2) with test functions in AE (Ω) instead of AE (Ω), because there is no contribution to the functional from the boundary of Ω. Problem (6)  give just a brief analysis of this formulation and possible solutions.

Symmetries of the Laplace Equation
One of the most efficient approaches to tackling Laplace's equation is exploiting its symmetries. This equation is known to accommodate the Lie groups of conformal transformations on ℝ , which are precisely the non-degenerate symmetry Lie groups of invariance transformations which preserve angles between vectors in their domains. Any such group can be decomposed into one-parameter subgroups. Each member É Ê of a conformal one-parameter Lie group 9É Ê ; Ê∈ℝ can be seen as a re-parametrization of ℝ ; In this event, When É Ê is linear, its action can be faithfully represented by an appropriate non-degenerate linear map Ð: ℝ ⟶ ℝ , meaning É Ê ( ) = Ð ∀ ∈ ℝ . Specifically, any transformation represented by a subgroup of the orthogonal group ¾(ƒ) ≔ 9Ð ∈ ℳ × (ℝ): Ð C = Ð ¤ ; exhibits the required properties and these suffice for simplifying equation (6), provided that they leave the boundary condition invariant. Except for the case on ƒ = 2, ¾(ƒ) has infinitely many one-parameter subgroups.
For an illustration of this example, we will consider a simple solution of Laplace's equation, using the ƒ −ball of radius ±; Ω = . (0, ±) for convenience, with the boundary constraint Ò 2| |2 on $Ω. Consider Ò to be a continuously differentiable real-valued function defined on an open subinterval of the reals containing 9±;. In this setting, any element of ¾(ƒ) leaves equation (6) invariant, and we can effectively deduce the Lie group invariant b( ) = || || which eliminates the group We hereby seek a solution of functional form Y_ ( ) = Ó(b).
where ∈ ℝ is a constant of integration. For a further constant Ž of integration, we have the solutions: for ≠ 0. Due to uniqueness of the weak solution, it becomes clear that (à) often lacks a classical solution for this case, taking Ž to be zero and the function Ò to be the identity for instance, recalling ℎ( ) = Ò(2| |2 on $Ω. This is because Y_ ( ) as computed is not continuously extendable at = 0 ∈ . (0, ±) . Nevertheless, the above solutions Y_ ( ) are harmonic functions on . (0, ±)\90; and details about such functions are seen in classical potential theory.
The solutions derived above to both (3) and (6) have been obtained after placing certain restrictions in either case. It is useful to determine characteristics of these two differential equations in other settings. In particular, Laplace's equation (6) can be restructured in several ways after implementing its potential transforms from conservation laws [5] to give optional vantage points for determination of subgroups of the overall admitted Lie symmetry group. This is instrumental in solving for group invariant solutions to (6), other than the fundamental solutions given here.

Conclusion
In many everyday applications, analogous quantified functionals are not differentiable, unlike the continuously differentiable examples considered above. In order to embrace a broader scope of functionals, we test instead for their lower semi-continuity and convexity. Given a functional : W → ℝ ∪ 9+∞; the domain â of is the active region for our analysis; â = 9Y ∈ W: (Y) ∈ ℝ; is lower semi-continuous at Y â if for every sequence {Y } ℕ â which converges in norm to Y , we have liminf →ã (Y ) ≥ (Y ) . is lower semi-continuous on â if it is lower semi-continuous at every Y ∈ â. is said to be convex if If is both convex and lower semi-continuous but not differentiable, then is still termed subdifferentiable and we invoke the Fenchel subdifferential of at ¡ ∈ â: $ (¡) = 9Y * ∈ W * : EY * , Y − ¡F ≤ (Y) − (¡) ∀Y ∈ W; We have the subdifferential $ (¡) to be nonempty upon subdifferentiablity of on â and elements of $ (¡) are called subgradients of at ¡. $ (¡) is always a convex set. If is in addition differentiable at ¡, then $ (¡) is a singleton which coincides with the classical differential of at ¡; $ (¡) ∈ W * , where W * is the dual of W. A necessary and sufficient condition for Y_ to be a local minimizer of is that 0 ∈ $ (Y_ ), and this is how we initiate weak formulations in this setting. As we have done in our included illustrations, we may first consider the problem set in a larger reflexive space to make arguments about existence and uniqueness of solutions using the same given optimization theorem in the introduction.