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Questions tagged [coercive]

For specific questions related to properties of coercive functions. In particular, these are commonly used in the optimization community.

In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the context, different exact definitions of this idea are in use.

An (extended-real-valued) function $$f: \mathbb{R}^n \to \mathbb{R} \cup \{ - \infty, + \infty \}$$ is called coercive iff $$f(x)\to +\infty {\mbox{ as }}\|x\|\to +\infty .$$

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On coercivity and compactness

I need to prove the following: Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be continuous on all of $\mathbb{R}^n$. $f$ is coercive $\iff \forall \alpha\in\mathbb{R}.\left\{x \mid f(x) \leq \alpha \right\}$ is compact. you can assume for the…
RDizzl3
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Proving that a strongly convex function is coercive

I am having trouble with this proof. I am given the following 2 definitions: 1) A function $f$ is coercive if $\lim_{||x|| \rightarrow \infty} f(x) = \infty$ 2) A $C^2$ function $f$ is strongly convex if there exists a constant $c_0 > 0$ such that:…
r.emp
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coercive implies bounded, variational principle

I have a question about the proof of the variational principle, see below. Any help is much appreciated! How does it follow from coercivity that $(x_k)_k$ is bounded? Why is $\alpha_0 > - \infty$? $\,$ Prerequisites Let $(X, || · ||_X )$ a normed…
augustin souchy
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How to determine coercive functions

A continuous function $f(x)$ that is defined on $R^n$ is called coercive if $\lim\limits_{\Vert x \Vert \rightarrow \infty} f(x)=+ \infty$. I am finding it difficult to understand how the norm of these functions are computed in order to show that…
clarkson
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Definitions of coercivity - functional analysis

In my PDE courses I've come across two different definitions or coercivity of a functional $\mathit{F}: \mathit{H} \rightarrow \mathbb{R}$ where $\mathit{H}$ is a Hilbert space. Definition 1: For the product space $\mathit{H} \times \mathit{H}$ for…
Len
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Is the bilinear form $A(\cdot,\cdot)$ $l_{2}$-elliptic or coercive?

Consider the space $l_{2} = \{ (x_{n})_{n\in \mathbb{N}} \subset \mathbb{R} : \sum_{n=1}^{\infty} |x_{n}|^{2} < \infty \}$. Now, consider the bilinear form $A(\cdot,\cdot):l_{2}\times l_{2}\to \mathbb{R}$, which is given by \begin{equation} A(x,y)…
tnt235711
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Functions that are coercive along every line

A function $f:\mathbb{R}^n\to\mathbb{R}$ is called coercive if $$ \lim_{\|x\|\to\infty}f(x)=\infty. $$ To show that $f$ is coercive, we need to prove that for every sequence $\{x_n\}$ with $\|x_n\|\to\infty$, it holds that $f(x_n)\to\infty$. My…
Kittayo
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Coercive/(weakly) semicontinuous function: extreme values

Consider functionals of the form $$\phi : X \rightarrow \mathbb{R} \cup\{+\infty\},$$ where $X$ is an arbitrary, normed vector space. In particular, $X$ may be of infinite dimension. I would be fine with restrictions like Banach-spaces or…
welahi
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Coercivity and spectral gap: understanding the equivalence

I am referring to this paper, p. 21. First, there is the following definition of coercivity: Let $L$ be an unbounded operator on a Hilbert space $\mathcal{H}$ with kernel $\mathcal{K}$ and let $\tilde{\mathcal{H}}$ be another Hilbert space…
selector
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In Calculus of Variation: Problem applying variational principle theorem

Let $f:\mathbb R^m \rightarrow [0,+\infty)\;$ be a smooth function that vanishes on a finite set $A\;$ where $\vert A \vert\; \ge 2$ and the maps $v:(l^{-},l^{+}) \rightarrow \mathbb R^m\;$ defined by $\mathcal M= \{\;v\in W^{1,2}_{loc}…
kaithkolesidou
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Coercivity of an integral operator in $L^2$-norm

Let us consider the integral operator $T:L^2(0,1)\to [0,\infty)$ such that for all $k\in L^2(0,1)$, $$ T(k)=\int_0^1 k_t^2e^{-2 \int_0^t k_s d s} d t. $$ Is the operator $T$ coercive in the $L^2$ sense, i.e., $ T(k)\to \infty$ as $|k|_{L^2}\to…
John
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Show that $f(x_1,x_2)=2x_1+(x_2-x_1^2)^2+(1-x_1)^2$ is coercive

I am trying to show that the function $$f(x_1,x_2)=2x_1+(x_2-x_1^2)^2+(1-x_1)^2$$ is coercive on $\mathbb{R}^2$. To show the function is coercive, we require $\|(x_1,x_2)\|\rightarrow+\infty\implies f(x_1,x_2)\rightarrow +\infty.$ We proceed by…
M B
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A strictly convex polynomial is coercive if and only if it has a positive definite Hessian

I have some difficulties in the following problem. Thank you for all comments and helping. Let $f:\mathbb{R}^n\rightarrow \mathbb{R} (n\in \mathbb{N})$ be a polynomial. Suppose that $f$ is strictly convex, i.e., for all $x,y \in\mathbb{R}^n,…
blindman
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Variational formulation, coercivity of: $ a(v,v) = \int_{\Omega} (\Delta v)^2 + \int_{\partial\Omega} v^2$

i'm trying to solve the Poisson equation: $$ \begin{split} -\Delta u &= f \quad \text{in } \Omega\\ u &= g \quad \text{on } \partial \Omega, \end{split} $$ where $ \Omega$ is bounded a Lipschitz-continues domain and $u \in H^2(\Omega), f,g \in…
skymath
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Self-adjoint operators and index of quadratic form associated to it.

Let $B$ a bounded self-adjoint operator on a Hilbert space $H=\{f: L^2(\mathbb{R}^n)\rightarrow \mathbb{R}\}$ with an associated inner product $(\cdot,\cdot).$ Take $V=span\{f_1, f_2, \ldots, f_n\}$ a finite dimensional subspace of $H$. Prove that…
Frank Zermelo
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