# Strict monotonicity and unique continuation for general non

We study some basic properties of nonlinear Kato class and respectively, for Also, we study the problem in where is a bounded domain in and the weight function is assumed to be not equivalent to zero and lies inin the case where.

Finally, we establish the strong unique continuation property of the eigenfunction for the -Laplacian operator in the case where. The Kato class was introduced and studied by Aizenman and Simon see [ 1 ].

Forit consists of locally integrable functions onsuch that Forthe following classes were defined by Zamboni see [ 2 ] : the class of functionssuch that and the class of functions such that and. Section 2 of the present paper is devoted to the study of some basic properties of the nonlinear Kato class andrespectively, for. Among other things, we show that the Lorentz space is embedded into see Lemma 10 as well as that is a complete topological vector space see Remark 11 and Lemma The -Laplacian operator is a generalization of the Laplace operator, where is allowed to range over ; in our case whereit is written as where andwith bounded domain in.

We are concerned with the following problem: and the weight function is assumed to be not equivalent to zero and lies in in the case. Specifically, we are interested in studying a family of functions which enjoys the strong unique continuation property, that is, functions besides the possible zero functions which have zero of infinite order. Definition 1. We say that a function vanishes of infinite order at point if for any natural number there exists a constantsuch that for all and for small positive number.

Definition 2. We say that 6 has a strong unique continuation property if and only if any solution of 6 in is identically zero in provided that vanishes of infinite order at a point in. There is an extensive literature on unique continuation. The same work is done by Chiarenza and Frasca, but for linear elliptic operator in the case where when [ 5 ]. Let us recall some known results concerning Fefferman's inequality as follows:.

In [ 6 ], de Figueiredo and Gossez prove 9 in the case whereassuming that with.

## Non-linear equation, numerical methods

Later in [ 7 ], Jerison and Kening showed the same result taking in the Stummel-Kato class. We point out that it is not possible to compare the assumptions and.

Chiarenza and Frasca [ 5 ] generalized Fefferman's result proving 9 under the assumption that with and. In [ 3 ], Schechter gave a new proof of 9 assuming that. In this section, we gather definitions and notations that will be used throughout the paper. We also include several simple lemmas. Bywe will denote the space of functions which are locally integrable onand by the space of functionssuch that.

Definition 3. For any andwe set where. We say that belongs to the space if for all. Definition 4.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

It only takes a minute to sign up. But my doubt now follows: if you assume strict convexity any linear combination with a different bundle in the same indifference curve will be strictly preferable for a preference, it seems to me that any degree of monotonicity will be strict -- right?

Am I missing something? The difference is that strong monotonicity requires only one more of any good to be strictly preferred. Consider preferences like those depicted below. The top indifference curve U1 is strictly less preferred than the bottom U2. Preferences over these goods which might better be called "bads" are strictly convex.

However, these preferences are in no degree monotonic. In a sense, they are monotonically decreasing, but we usually take "monotonic preferences" to be increasing as in the definition I gave above. For the right domain, preferences of this kind are indeed complete and transitive. So, strictly convex preferences do not imply monotonicity at least in the direction that you want.

Then, it is true that this form of monotonicity together with strict convexity as defined above imply strong monotonicity as defined above. This is a contradiction. Thus, the preferences are strongly monotonic.

Here is an attempt to make up for my ugly mistake on the strict convexity of Leontieff preferences. If I read you right, the guess you want to check is whether for complete, transitive and continuous preferences, strict convexity implies that any monotonic preference relation is also strictly monotonic.

I believe your guess is right and this can be shown by contradiction the construction of the different bundle in the proof are illustrated in the picture. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 6 years, 7 months ago. Active 6 years, 6 months ago. Viewed 10k times.

John Doe John Doe 3 3 silver badges 10 10 bronze badges. The other properties are assumed so that we can derive "demand functions. Kreps' textbook calls the same property "strict monotonicity".By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

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However, another definition of strict monotonicity says that marginal utility of each good should be strictly positive. Is there a way to resolve this? Leontief preferences are the usual example for weakly but not strongly monotonic preferences.

The indifference curve passing through 0,0 is L-shaped for both these and for Cobb-Douglas preferences. However for the Cobb-Douglas case you can prove that the optimal choice of the consumer assuming positive income is never on the boundary, as that yields the lowest possible utility. After this, assuming the consumer makes optimal choices, the utility function is strongly monotonic in the local environment of her choice. They do not satisfy either condition. The condition that all marginal utilities must be positive is inherently problematic because it does depend not just on the underlying preferences.

For one, not every utility representation needs to be differentiable. But even that is not enough. Suppose there is a single good and more is better than less.

The problem just mentioned is actually even worse. Long story short: Every partial derivative being strictly positive is a sufficient but not necessary condition for a differentiable function to be increasing in every coordinate.

Yet another issue is that it is not entirely clear how one defines the derivative at the boundary of the commodity space. There are different notions of differentiability which need not be equivalent. This is easily seen from Giskard's answer: as he rightly points out, the log transformation of the CD-utility is not defined when either coordinate is 0. In such situations, you should derive using first principles.

The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Are Cobb-Douglas preferences monotone according to the marginal utility condition? Ask Question. Asked 3 months ago. Active 3 months ago. Viewed times. PGupta PGupta 8 8 bronze badges. I have seen strong and strict being used interchangeably. Active Oldest Votes. Giskard Giskard Is this a general rule or specific to this utility function?

My exact statement is " Cobb-Douglas preferences are strongly monotonic over the positive part ".Full-text: Access denied no subscription detected However, an active subscription may be available with MSP at msp. We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription.

Read more about accessing full-text. We show a monotonicity relation between the scattering coefficient q and the local Neumann-to-Dirichlet operator that holds up to finitely many eigenvalues. Combining this with the method of localized potentials, or Runge approximation, adapted to the case where finitely many constraints are present, we derive a constructive monotonicity-based characterization of scatterers from partial boundary data.

Source Anal. PDEVolume 12, Number 7 Zentralblatt MATH identifier Monotonicity and local uniqueness for the Helmholtz equation. PDE 12no. Read more about accessing full-text Buy article. Article information Source Anal. Export citation. Export Cancel. References G.

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Differential Equations 84 :2— Zentralblatt MATH: You have access to this content. You have partial access to this content. You do not have access to this content. More like this.We will study the spectrum for the biharmonic operator involving the laplacian and the gradient of the laplacian with weight, which we call third-order spectrum.

We will show that the strict monotonicity of the eigenvalues of the operatorwhereholds if some unique continuation property is satisfied by the corresponding eigenfunctions. Based on the works of Anane et al. This spectrum is defined to be the set of couples such that the problem has a nontrivial solution. This spectrum, which is denoted byis an infinite sequence of eigensurfacessee Section 3. Whenthe zero-order spectrum is defined to be the set of eigenvalues such that the problem has a nontrivial solution.

In this case the spectrum is denoted by. The eigenvalue problem 1. Definition 1. We say that solutions of problem 1. Pif the unique solution which vanishes on a set of positive measure in is. In the literature there exist several works on unique continuation.

We refer to the works of Jerison and Kenig [ 4 ] and Garofalo and Lin [ 5 ], among others. The unique continuation property as defined above differs from the usual notions of unique continuation, see [ 6 ] for more details. We say that is strict monotone with respect to the weight iffor all. Here we use the notation to mean inequality almost everywhere together with strict inequality on a set of positive measure.

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Since the pioneer works of Carleman [ 7 ] in on the unique continuation, this notion has been the interest of many researchers in partial differential equations, see for instance [ 458 ]. Inde Figueiredo and Gossez [ 6 ] proved that strict monotonicity holds if and only if some unique continuation property is satisfied by the corresponding eigenfunction of a uniformly elliptic operator of the second order.

InGossez and Loulit [ 8 ] have proved the unique continuation property in the linear case of the laplacian operator. The unique continuation property of the biharmonic operator was proved recently by Cuccu and Porru [ 9 ]. Our purpose in the fourth section is to show that strict monotonicity of eigensurfaces for problem 1. Let be a finite dimensional separable Hilbert space. We denote by and the inner product and the norm of the spacerespectively. Let be a compact operator.

Lemma 2. All nonzero eigenvalues of the operator are obtained by the following characterizations: where denotes the class of -dimensional subspaces of. Moreover, zero is the only accumulation point of the set of all eigenvalues of. Here, the eigenvalues are repeated with its order of multiplicity, and the eigenfunctions are mutually orthogonal [ 10 ].

We define the third-order eigenvalue problem of the biharmonic operator as follows:. If is a solution of 3. Lemma 3.By a non-linear equation one means see  —  an algebraic or transcendental equation of the form. Equation 1 and system 2 can be treated as a non-linear operator equation:.

The most important iteration methods for the approximate solution of 3both of a general and of a special form, characteristic for discrete grid methods for solving boundary value problems for equations and systems of partial differential equations of strongly-elliptic type, are the object of study of the present article. Non-linear operator equations connected with the discussion of infinite-dimensional spaces see, for example  —  are a very broad mathematical concept, including as special cases, for example, non-linear integral equations and non-linear boundary value problems.

Numerical methods for the approximate solution of them include also methods for their approximation by finite-dimensional equations; these methods are treated separately.

One of the most important methods for solving an equation 3 is the simple iteration method successive substitutionwhich assumes that one can replace 3 by an equivalent system. Then by a general contractive-mapping principle see  —  and Contracting-mapping principle equation 1 has a unique solution, the method of simple iteration.

Under certain additional conditions the Newton—Kantorovich method leads to an error estimate of the form. At every iteration of this method one has to solve the system of linear algebraic equations with matrix. The Newton—Kantorovich method belongs to the group of linearization methods 3.

Another method in this group is the secant method. Among the several versions of descent methods one can mention the methods of coordinate-wise descent, several gradient methods and, in particular, the method of steepest descent, the method of conjugate gradients, and others, and also their modifications see  — . A number of iteration methods for the solution of the equations 3describing a certain stationary state, can be treated as discretizations of the corresponding non-stationary problems.

Therefore, methods of this class are called stabilization methods cf. Adjustment method and, for example, . Examples of such non-stationary problems are problems described by a system of ordinary differential equations. The introduction of an additional independent variable is characteristic for the method of differentiation with respect to a parameter a continuation method, see . Generally speaking, the system 9 determines. In the method of continuation with respect to a parameter cf. Both these methods are in essence iteration methods for solving 2 with a special procedure for finding a good initial approximation.

In the case of systems the problem of localizing the solution causes great difficulty. Since the majority of iteration methods converge only in the presence of a fairly good approximation to a solution, the two methods described above often make it possible to dispense with the need to localize a solution directly.

For localization one also frequently uses theorems based on topological principles and on the monotonicity of operators see  — . For the solution of equations 1which are the simplest special cases of 3the number of known iteration methods applicable in practice is very large see, for example,  — .

Apart from the ones already considered one can mention, for example, the iteration methods of higher order see which include Newton's methods as a special case, and the many iteration methods especially oriented to finding real or complex roots of polynomials. Iteration methods for solving equations 3 that arise in grid analogues of non-linear boundary value problems for partial differential equations are special cases of methods for solving grid systems see, for example,  — .By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. But my doubt now follows: if you assume strict convexity any linear combination with a different bundle in the same indifference curve will be strictly preferable for a preference, it seems to me that any degree of monotonicity will be strict -- right?

Am I missing something? The difference is that strong monotonicity requires only one more of any good to be strictly preferred. Consider preferences like those depicted below.

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The top indifference curve U1 is strictly less preferred than the bottom U2. Preferences over these goods which might better be called "bads" are strictly convex. However, these preferences are in no degree monotonic. In a sense, they are monotonically decreasing, but we usually take "monotonic preferences" to be increasing as in the definition I gave above. For the right domain, preferences of this kind are indeed complete and transitive. So, strictly convex preferences do not imply monotonicity at least in the direction that you want.

Then, it is true that this form of monotonicity together with strict convexity as defined above imply strong monotonicity as defined above. This is a contradiction. Thus, the preferences are strongly monotonic. Here is an attempt to make up for my ugly mistake on the strict convexity of Leontieff preferences. If I read you right, the guess you want to check is whether for complete, transitive and continuous preferences, strict convexity implies that any monotonic preference relation is also strictly monotonic.

I believe your guess is right and this can be shown by contradiction the construction of the different bundle in the proof are illustrated in the picture.

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Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 6 years, 7 months ago. Active 6 years, 6 months ago. Viewed 10k times. John Doe John Doe 3 3 silver badges 10 10 bronze badges.

The other properties are assumed so that we can derive "demand functions. Kreps' textbook calls the same property "strict monotonicity". Jehle and Reny on the other hand agree with you definition of strict monotonicity. Finally I have seen papers referring to what you call "strict monotonicity" as "monotonicity".

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Quite a mess It is correct, however, in my solution below. However, I think there is definitely agreement between strict and strong, as the same concepts appear elsewhere and more generally in mathematics. Check me if I'm wrong. See wikipedia. I believe the idea is that strongly convex means that strictly increasing the quantity of at least one good strictly increases utility. Strict convexity means all goods must be increased to guarantee strictly more utility.