Y. Giga, S. Goto, H. Ishii, and M.-H. Sato. Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Ind. Math. J., 40(2):443--470, 1991. 97  Home/Search   Document Not in Database   Summary   Related Articles   Check

....hold) we prove that there exists a unique solution u of (4) which is convex in the space variable at each time, and this for any convex initial data u 0 without any restriction on its growth at in nity. The proof relies strongly on the convexity preserving property of Giga, Goto, Ishii and Sato [21] that we extend to our more singular case. Compared to the result we previously obtained in [7] by working directly on (1) we do not assume anymore u 0 to be coercive and we extend the result to equations like (4) For completeness, we conclude this introduction by describing the results we ....

.... The geometrical equation: the classical framework A priori the nonlinearity F is discontinuous on D (see (7) In this section, we provide assumptions on b ensuring that we are in the classical framework, which means that the (classical) level set approach applies readily to (6) see [18] 13] [21] and [10] In this classical framework, F has to be continuous, except at p = 0: The typical example is the mean curvature equation (see Example 3.1) More precisely, we start by recalling the assumptions as they appears in [21] In the sequel, k k is any norm on SN and S = f 2 IR : j j ....

[Article contains additional citation context not shown here]

Y. Giga, S. Goto, H. Ishii, and M.-H. Sato. Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J., 40(2):443-470, 1991.

.... for quasilinear equations i.e. for equations involving the above mentioned diculty on the Du dependence concern only uniformly continuous viscosity solutions : we refer the reader the Users guide of viscosity solutions of Crandall, Ishii and Lions [9] and to Giga, Goto, Ishii and Sato [15] for results in this direction. The aim of this article is to push as far as possible the classical arguments used for proving comparison results for viscosity solutions in order to obtain such results for the largest possible class of quasilinear equations and initial datas. In the general ....

Y. Giga, S. Goto, H. Ishii, and M.-H. Sato. Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J., 40(2):443-470, 1991.

....singularities provided no fattening occurs. The proof is based, as in [3] 9] on constructing viscosity supersolutions to the reaction diffusion PDE in terms of the signed distance function d to suitable level sets of the viscosity solution to the generalized geometric motion (1. 2) 7] 10] [12]. The novelties here are the presence of obstacles, that entail lack of regularity of u even for smooth initial data, and the use of an explicit traveling wave dictated by formal asymptotics. Inspired by [19] the first issue is tackled with a suitable notion of viscosity supersolution and ....

....is positive outside Sigma 0 . Let denote the (continuous) viscosity solution of the nonlinear degenerate parabolic PDE (2. 3) t Gamma i ffi ij Gamma x i x j jr j 2 j x i x j Gamma gjr j = 0 in R n Theta (0; T ) satisfying ( Delta; 0) d 0 ( Delta) 7] 8] 10] [12]. Such an expression says that level sets of evolve formally in their normal direction with velocity V = t jr j = g, as stated in (1.2) Let Sigma(t) indicate the zero level set of . Since Sigma(t) is independent of the special form of ( Delta; 0) provided f ( Delta; 0) 0g = ....

[Article contains additional citation context not shown here]

Y. Giga, S. Goto, H. Ishii, and M.H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40 (1991), pp. 443--470.

....direction according to (1.1) 31] Such a w is thus the unique viscosity solution to the degenerate nonlinear PDE (1. 2) t w Gamma jrwj div Gamma rw jrwj Delta Gamma jrwjg = 0 in R n Theta (0; T ) satisfying the initial condition w( Delta; 0) w 0 ( Delta) 2 C(R n ) 7] 10] 13] [17]. The zero level set of w 0 coincides with Sigma 0 and w 0 is constant outside a large ball; thus w(x; t) turns out to be constant for jxj large [13] If d 0 denotes the signed distance to Sigma 0 which is negative in I(0) the inside of Sigma 0 , then a typical choice of w 0 is d 0 truncated ....

....for as long as I(0) fx 2 R n : w 0 (x) 0g, 1.3) Sigma(t) fx 2 R n : w(x; t) 0g; Sigma : f(x; t) 2 R n Theta [0; T ] x 2 Sigma(t)g) is then called generalized geometric evolution and coincides with the classical motion (1. 1) before the onset of singularities [2] 7] 13] [17]. The notions of inside I(t) and outside O(t) of Sigma(t) are thus meaningful in terms of w: I(t) fx 2 R n : w(x; t) 0g; O(t) fx 2 R n : w(x; t) 0g: The flow (1.3) is unique and is well defined even past singularities. These are typically isolated points (x; t) that correspond to ....

[Article contains additional citation context not shown here]

Y. Giga, S. Goto, H. Ishii, and M.H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40 (1991), pp. 443--470.

....ingredients are also discussed. 1. Introduction Let Sigma(t) be the flow that emanates from a smooth surface Sigma 0 of codimension 1 in R n and propagates in its normal direction with velocity V satisfying the following geometric law (1. 1) V = g; in the viscosity sense [3] 10] 16] [18]. Hereafter denotes the sum of the principal curvatures of Sigma(t) and g is a given function. This purely geometric evolution may exhibit singularities, topological changes, and even nonuniqueness [2] 4] 38] Besides its mathematical interest, 1.1) may be viewed as a scaled version of the ....

.... Gamma rw jrwj Delta Gamma g = 0; w( Delta; 0) w 0 ( Delta) with w 0 satisfying Sigma 0 = fx 2 R n : w 0 (x) 0g [32] Equation (1.3) corresponds to the motion of all level sets of w( Delta; t) according to (1. 1) and must be interpreted in the viscosity sense [3] 10] 13] 16] [18]. The dimension has thus been increased by one in replacing Sigma(t) by w( Delta; t) Our method being local thus represents a valid alternative to the global level set approach. Let U ;h; and Sigma ;h; indicate the fully discrete solution and its zero level set Sigma ;h; t) fx 2 Omega ....

[Article contains additional citation context not shown here]

Y. Giga, S. Goto, H. Ishii, and M.H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J. 40 (1991), 443--470.

....commutes with translations and additions of constants (which yields an x and u independent F ) In this context, a comparison result holds for the viscosity solutions of (1) in BUC(IR N ) see for example M.G Crandall, I. Ishii, P .L. Lions [6] or Y. Giga, S. Goto, I. Ishii and M. H Sato [8]) Here, on the contrary, the answer is no in general, even for a linear semigroup if the assumption of commutation with translations is removed. In Section 5, we build an example of a semigroup de ned on BUC(IR N ) which satisfy the assumptions of Section 2 and which is associated to a ....

Y. Giga, S.Goto, H.Ishii and M.-H. Sato (1991). Comparison Principle and Convexity Preserving Properties for singular Parabolic Equations on Unbounded Domains. Indiana University Mathematics Journal 40, No. 2.

....and the lower semi continuous envelope v (O; 0) 0. In particular, v (x; 0) u(x; 0) on Omega Gamma 2 . Further, u(x; t) 0 for all x 2 t) so v u on the parabolic boundary f(x; t) 2 Omega j 0 t ffig. It follows from 14 the comparison principle that v u everywhere on Omega (see [GGIS], p. 463) In particular, u(x; t) 0 for all x 2 Gamma a 0 (t) 0 t ffi. Let D(t) be the bounded open set in IR n whose boundary consists of portions of Sigma (t) Sigma Gamma (t) and the surface of revolution Gamma a 0 (t) for each 0 t ffi. Write D = f(x; t) j x 2 D(t) 0 ....

.... t) 0 v a (x; t) u(x; t) on ( B n Delta Omega Gamma282 Theta f0g) 2 4 [ 0t ffi 0 [B n Delta Omega Gamma t) Theta ftg 3 5 : It follows from the comparison principle that 0 v a (x; t) u(x; t) for all (x; t) 2 S 0t ffi 0 [B n Delta Omega Gamma t) Theta ftg (see [GGIS], p. 463) Moreover, since a Delta 2 b by the assumption on a, which implies p a 2b Delta, we can define ffi 0 by ffi : min ae a C Delta Gamma r a 2b ; ffi 0 oe (4.16) where ffi 0 is defined in (4.13) Then Ct a r a 2b Delta whenever 0 t ....

Y. Giga, S. Goto, H. Ishii and M. H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J. 40 (1991), 443--470.

.... t t 0 and e t t 0 . To this end, let us consider the function F : IR N (0; 1) IR N n f0g SN IR de ned by (11) and let us observe that, under the assumptions (A1) A2) and (H1) H4) the viscosity solution theory for fully nonlinear degenerate parabolic equations (see [3, 6, 7, 10]) provides, for every uniformly continuous function u 0 : IR N IR, the existence of a unique solution u 2 C(IR N [0; 1) of the initial value problem ( u t F (x; t; Du; D 2 u) 0 in IR N (0; 1) u(x; 0) u 0 (x) in IR N : 16) Moreover, since the function F is geometric, ....

Y. Giga, S. Goto, H. Ishii, M.H. Sato, \Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains", Ind. Math. J. 40 (1991), 443-470.

....conditions. We first recall that this approach was first introduced by Evans and Spruck[9] for defining a weak notion of motion by mean curvature without constraints i.e. in IR N and then extended to more general types of motions by Chen, Giga and Goto [6] See also Giga, Goto, Ishii and Sato[11]) We consider here fully non linear degenerate parabolic pdes of the following form u t F (x; t; Du;D 2 u) 0 in Omega Theta (0; 1) 28) to gether with some initial condition u(x; 0) u 0 (x) in Omega ; 29) and with a nonlinear Neumann boundary condition L(x; t; Du) 0 on ....

Y. Giga, S. Goto, H. Ishii & M.H. Sato , Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Ind. Univ. Math. J. Vol 40. n 2 (1991).

.... t H(t; x; y h; v x ; v y ) 0 at any local minimum of h v for any C 1 test function v: Thus h is a viscosity super solution of equation (7) If h (x; y; 0) x; y; 0) 0 for all x and y; then h (x; t) x; t) 0 by the comparison principle (the reader is referred to [11][20] for the proof) This basically says that if y (x; y; t = 0) 0 initially, then y (x; y; t) 0 for all time It also implies that f = cg will remain as a graph throughout the evolution. Therefore, we can remove the sign( y ) term from the derived level set equation (2) of this class of ....

Y. Giga, S. Goto, H. Ishii, and M.-H. Sato. Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J., 40(2):443--470, 1991.

No context found.

Y. Giga, S. Goto, H. Ishii, and M.-H. Sato. Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Ind. Math. J., 40(2):443--470, 1991. 97

No context found.

Y. Giga, S. Goto, H. Ishii, and M.H. Sato, Comparison principle and convexity preserving properties for singular degenerated parabolic equations on unbounded domains, Indiana Univ. J., Vol. 40 (1991), 443-470.

No context found.

Y. Giga, S. Goto, H. Ishii, and M.H. Sato, Comparison principle and convexity preserving properties for singular degenerated parabolic equations on unbounded domains, Indiana Univ. J., Vol. 40 (1991), 443-470.

No context found.

Y. Giga, S. Goto, H. Ishii, and M.-H. Sato. Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J., 40(2):443--470, 1991. 24

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC