A DUALITY APPROXIMATION OF SOME NONLINEAR PDE s

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1 ISSN Ann. Acad. Rom. Sci. Ser. Math. Appl. Vol. 8, No. /206 A DUALITY APPROXIMATION OF SOME NONLINEAR PDE s Dan Tiba Abstract We discuss a discretization approach for the p - Laplacian equation and a variational inequality associated to fourth order elliptic operators, via a meshless approach based on duality theory. MSC: 35J20, 35J25, 65N99. keywords: nonlinear elliptic equations, Fenchel theorem, approximation, meshless method. Introduction We consider the well known p - Laplacian boundary value problem, expressed in the variational form (minimization of energy) via the Dirichlet principle: Min [ y p + y p ]dx fydx, () y W,p 0 () p where R d is a bounded domain and p > d 2 is given, f L q () with p + q =. The space W,p 0 () is the usual Sobolev space. Accepted for publication on April 8-th 206 dan.tiba@imar.ro, Institute of Mathematics, Romanian Academy, Bucharest and Academy of Romanian Scientists; 68

2 A duality method 69 The existence of a unique weak solution y W,p 0 () is standard due to the coercivity and strict convexity of the functional (). It may be interpreted as solving the nonlinear boundary value problem d ( ) p y y + y p 2 y = f in, (2) x i x i i= y = 0 on, (3) in a weak sense in W,p 0 (). The basic idea of our approach is to interprete the boundary condition in (2), (3) as a constraint in the minimization problem (), as it is in fact. These infinitely many pointwise constraints make sense due to the Sobolev embedding theorem W,p 0 () C(), if p > d. The method that we introduce in this note, discretizes (3) without using a discretization of or, it is a meshless method. Moreover, we are not applying the (rather complex) bases (like RBF for instance) that are usual in meshless methods [2], [0]. The idea goes back to [6], [], [9], Ch. 6, where the Kirchhoff - Love model for arches is discussed. Since this model involves just a finite number of boundary conditions, then the explicit solution is obtained [6]. As a comparison, underlying the efficiency of our approach, in [3], [4] finite element discretizations and the locking problem are discussed for the Kirchhoff - Love arches. In Section 2, we introduce the approximating problem and prove some convergence properties. Its solving is discussed in Section 3. In the recent papers [7], [8] similar ideas were used for the distributed obstacle problem in elliptic equations, including numerical experiments. In Section 4, we briefly analyze an elliptic variational inequality for fourth order operators, via the same technique. 2 Approximation Let x i i N be a sequence of points dense in. No elements are used, no uniformity or regularity assumptions are made for the geometry and its approximation. The approximation of () is given by [ y p + y p ]dx fydx, (4) p Min y W,p () y(x i )=0, i=,n

3 70 Dan Tiba for any n N. We denote by (P n ) the problem (4). Proposition. The problem (P n ) has a unique solution y n W,p (). Moreover, we have y n y, the solution of (), strongly in W,p (). Proof. The existence and uniqueness of y n W,p () is again a consequence of the coercivity and strict convexity of the functional (4). We have the following inequality [ y n p dx + y n p ] fy n dx [ y p dx + y p ] fydx, (5) p p since y is admissible for (P n ), any n N. In particular, (5) shows that y n is bounded in W,p () due to the coercivity of (4). On a subsequence, denoted again by n, we may assume y n ỹ weakly in W,p (). Since the functional (4) is weakly lower semicontinuous, using (5) as well, we infer [ ỹ p dx + ỹ p ] fỹdx [ y p dx + y p ] fydx. (6) p p Notice that, for any i N, we have ỹ(x i ) = 0. This is satisfied by y n, n > i and remains valid by passing to the limit in W,p () weak (and implicitly in C()). As ỹ C() and x i is dense in, it yields ỹ W,p 0 (). Then, by (6) and the uniqueness of y, we obtain ỹ = y. The inequalities (5), (6) also show that [ y n p dx + y n p ] fy n dx [ y p dx + y p ] fydx, (7) p p for n. Since clearly fy n fy due to the weak convergence of y n, we obtain from (7) the convergence of the norms of the solutions and the strong convergence of y n, via a strong convergence criterion in uniformly convex Banach spaces. The convergence is valid on the initial sequence since the limit is unique. This ends the proof.

4 A duality method 7 Remark. This approximation argument can be easily extended to many elliptic boundary value problems, linear or nonlinear. The key point is to ensure the continuity of the solutions. To circumvent this argument, by supplementary approximation techniques or other procedures, would be of interest. 3 The duality argument We investigate now the solution of (4). It has a finite number of constraints and we shall compute the dual problem, using the Fenchel theory. The dual problem is finite dimensional and easier to solve. Define g n : W,p () ], + ] as 0 y(xi ) = 0, i =, n, g n (y) = + otherwise. And denote by h : W,p () R the continuous, concave mapping h(y) = [ y p + y p ] dx + fydx. (8) p Clearly, the problem (P n ) can be expressed as min g n(y) h(y), (9) y W,p () via (8), (8). Since g n is convex, lower semicontinuous and finite in certain points of W,p (), the Fenchel theorem [], allows to rewrite (9) as max (z) gn(z), (0) z W,p () h where h, g n denote concave/convex conjugates defined on the dual space W,p (). We compute now the conjugates from (0). Lemma. We have gn(z) 0, z = n α i δ xi, α i R, = i= +, otherwise, () where δ xi is the Dirac measure concentrated in x i.

5 72 Dan Tiba Proof. g n(z) = sup (z, w)w,p () W,p () g n (w) = w W,p () = sup w W,p () w(x i )=0, i=,n (z, w)w,p () W,p () 0. (2) If in ẑ W,p (), we have gn(ẑ) > 0, then we may change in (2) w λw with λ and infer that gn(ẑ) = +. Consequently, gn takes either the value 0 or +. If δ xi is the Dirac measure concentrated in x i, then δ xi W,p () since p > d and W,p () C(). By (2), we have gn(δ xi ) = 0, for i =, n. This remains valid for any z = n α i δ xi with α i R arbitrary scalars. Therefore, g n is null on this finite dimensional subspace in W,p (). Notice that: i= dom g n = z W,p () ; (z, w) W,p () W,p () = 0, w A n, (3) where A n = y W,p (); y(x i ) = 0, i =, n is a closed linear subspace in W,p (). n Denote B n = α i δ xi, α i R W,p (), a linear closed subspace. i= The polar space of B n is B n = y W,p (); (y, z) W,p () W,p () = 0; z B n = = y W,p (); y(x i ) = 0; i =, n = A n. Since by (3), dom g n is the polar of A n, the bipolar theorem [], p. 88, shows that dom g n = B n. This ends the proof. Proposition 2. The dual problem (0) is given by: n min q f z q ; z = α W,p () i δ xi, α i R. (4) i= It is a finite dimensional optimization problem.

6 A duality method 73 Proof. h (z) = inf (y, z)w y W,p (),p () W,p () h(y) = = inf (y, z f) W y W,p (),p () W,p () + p [ y p + y p ] q z f q, W,p () due to the elementary inequality ab p ap + q bq, with equality for a = b p. If we choose y W,p (), y = J(z f), then we get equality in (5) and, consequently, h (z) = q z f q W,p (). Above: J : W,p () W,p () is the duality mapping with weight e(α) = α p, α R. That is, it satisfies (see [5], p. 38): (J(z), z) W,p () W,p () = J(z) W,p () z W,p (), J(z) W,p () = e( z W,p () ). Here, we also use that W,p () is a reflexive Banach space and J has nonvoid values on W,p (). Together with Lemma, this ends the proof. Remark 2. If we denote by z n the unique solution of the dual problem (4), then the solution y n of the approximating problem (9) may be obtained from the relation z n f h(y n ), where h is the subdifferential of (8). See [], p. 88. Since z n is a linear combination of Dirac measures, the solution of the above relation is much simplified. If p = 2, it becomes linear and one can also get continuity in space by taking in problem (2) the right hand side f in L s (), s big enough, depending on dimension. Remark 3. A formal interpretation of the approximating problem (4) is as a mixed boundary value problem: pointwise Dirichlet conditions in x i, i =, n and Neumann conditions in the remaining of. 4 A fourth order variational inequality We formulate the variational problem

7 74 Dan Tiba K = Min y K 2 z H 2 () H0 (), y 2 dx hydx, (5) hzdx, (6) where h L 2 () is given and is a bounded domain. The problem (5), (6) can be interpreted as a simply supported plate and the unilateral condition in (6) is related to the mechanical work associated to the force h. It has a unique solution in K due to the coercivity and strict convexity of the functional (5). We approximate (5), (6) by an optimization problem with a finite number of constraints: K n = Inf y K n y 2 dx hydx, (7) 2 z H 2 (); z(x i ) = 0, i =, n, hzdx, (8) where x i i N is a dense subset as in the previous sections and K n is well defined if z is continuous, for instance if R 3, by the Sobolev theorem. Notice that the existence question in (7), (8) is not easy to be settled, due to the possible lack of coercivity. Under conditions in (8), the seminorm in (7) may not be a norm. However, the dual approximating problem (which is finite dimensional) has solutions and the Fenchel theorem may be applied. We fix: g : L 2 () R, g(z) = z 2 dx, (9) 2 which is a concave continuous mapping, which is a linear continuous operator, D : H 2 () L 2 (), Dy = y, (20) f n : H 2 () ], + ], (2) f n (z) = hz, z K n, + otherwise,

8 A duality method 75 a convex lower semicontinuous proper function. By (9) - (2), the problem (7), (8) may be rewritten as Inf f n(z) g(dz) (22) z H 2 () and the hypotheses of the Fenchel theorem are fulfilled (for z = 0, for instance). The conjugate g (z) = z 2 dx is clear and, for f 2 n, we state Lemma 2. We have f n(y) = 0, y K 0 n + h, + otherwise, where K 0 n H 2 () is the polar of the convex K n. Proof. f n(y) = sup z, y + h H 2 () H 2 () ; z K n 0. (23) If y has the form y = h + n α i δ xi, α i R and δ xi are Dirac measures i= from H 2 () concentrated in x i, then f n(y) = 0. Fix z 0 K n, such that z 0hdx =, which clearly exists. Then, K n z 0 is a cone and the unilateral condition can be equivalently expressed: h(z z 0 )dx 0. If y H 2 () is such that f n(y) > 0, we take z 0 + λ(z z 0 ) K n, λ > 0 and compute z 0 + λ(z z 0 ), y + h H 2 () H 2 () = + z 0, y H 2 () H 2 () + +λ z z 0, y + h as λ. (24) The value is impossible in (24), due to (23). It yields that f n(y) takes either 0, or + values, due to (23), (24). We also remark f n(y h) = sup z, y H 2 () H 2 () ; z K n

9 76 Dan Tiba for y K 0 n, i.e. f n(y h) = 0 for y h K 0 n. Outside K 0 n + h, we get greater than values, i.e. f n(y h) = +, y / K 0 n. This ends the proof. Remark 4. Denote by A n = Then y = h + n i= α i δ xi, α i R A 0 n = x H 2 (); x, z H 2 () H 2 (), z A n x H 2 (); y(x i ) = 0, i =, n, H 2 (). since, otherwise, we may take α i ± conveniently, and obtain a contradiction. Using again the definition of the dual and this property, we get A 0 n = y H 2 (); y, h H 2 () H 2 (), y(x i) = 0, i =, n = K n. It yields that K 0 n = A 00 n = conv (0 A n ), by the bipolar theorem, []. This further clarifies Lemma 2. By Fenchel Theorem, we infer: Proposition 3. We have Inf f n(z) g(dz) = Max z H 2 () y L 2 () g (y) fn(d y), where D : L 2 H 2 () is the adjoint operator of D. Remark 5. The dual problem reduces to the maximization of g under constraint D y K 0 n + h, a finite dimensional optimization problem. Acknowledgement This work was supported by CNCS Romania under Grant 02/20. References [] V. Barbu, Th. Precupanu. Convexity and optimization in Banach spaces. Sijthoff & Noordhoff, 978. [2] D.S. Broomhead, D. Lowe. Multivariable functional interpolation and adaptive networks. Complex Systems 2:32-355, 988.

10 A duality method 77 [3] D. Chenais, J.-C. Paumier. On the locking phenomenon for a class of elliptic problems. Numer. Math. 67: , 994. [4] D. Chenais, M. Zerner. Numerical methods for elliptic boundary value problems with singular dependence on a small parameter, necessary conditions. Comput. Meth. Appl. Mach. Engry. 5(-2):45-63, 994. [5] I. Ciorănescu. Aplicaţii de dualitate în analiza funcţională neliniară. Ed. Acad. RSR, 974. [6] A. Ignat, J. Sprekels, D. Tiba. Analysis and optimization of nonsmooth arches. SIAM J. Control Optim. 40(4):07-33, 200. [7] D.R. Merluşcă. A duality-type method for the obstacle problem. An. Şt. Univ. Ovidius, Constanţa 2(3):8-95, 203. [8] D.R. Merluşcă. A duality-type method for the fourth order obstacle problem. U.P.B. Sci. Bull., Ser. A 76(2):47-58, 204. [9] P. Neittaanmaki, J. Sprekels, D. Tiba. Optimization of elliptic systems. Theory and applications. Springer Monographs in Mathematics, Springer, New York, [0] M.J.D. Powell. Radial basis function approximations to polynomials. DAMPT preprint NA/6, 987. [] J. Sprekels, D. Tiba. Sur les arches lipschitziennes. C.R.A.S. Paris, Ser. I Math. 33(2):79-84, 2000.