Questions tagged [lipschitz-functions]

For question involving functions satisfying a Lipschitz continuity condition, that is, the distance ratio about the distance of $f(x)$ and $f(y)$ and that of $x$ and $y$ can be bounded independently of $x$ and $y$.

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Lipschitz Continuity of the Value Function [duplicate]

Suppose that $g:[0,1]^2 \rightarrow \mathbb{R}$ is a smooth function. Define the value function $$g^*(x) = \max_{t \in [0,1]} g(x,t).$$ Question: Under what conditions would $g^*$ be a Lipschitz ...
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Why is the Lipschitz Condition described using "Cones"?

I have often seen the Lipschitz Condition (Lipschitz Continuity) of mathematical functions being characterized through the following analogy: A function obeys the Lipschitz Condition if each point on ...
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Do we have any way of knowing if natural phenomena in the real world follow the "Lipschitz Condition"?

Recently, I keep coming across terms containing "Lipschitz" pertaining to statistical models and machine learning. This includes terms such as "p-lipschitz (rho), lipschitz convexity, ...
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When is a non-differentiable vector field a gradient field?

I have the following question regarding the connection between vector fields and gradient fields. Assume I have given a vector field $$\textbf{F} : \mathbb{R}^n \to \mathbb{R}^n.$$ I want to know when ...
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difference between lipschitz domain and domain satisfying the cone condition

I've come across two definition and I cant understand the differences between them. Maybe someone can help me..Whats the difference between a lipschitz domain and a domain satisfying the cone ...
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Does the following operation preserve Lipschitz continuity?

For fixed $r>0$, consider $\mathcal{L}: L^\infty(0,1) \rightarrow L^\infty(0,1)$, with $(\mathcal{L}f)(x) = \|f\|_{L^\infty(B_r(x))}$. If $f$ is Lipschitz continuous, is $\mathcal{L}f$ Lipschitz ...
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If $f$ is periodic and $C^1$ class, then Lipschitz continuous.

I proved if $f : \mathbb R\to \mathbb R$ is periodic and $C^1$ class, then $f$ is Lipschitz continuous. I wonder if my proof is correct and if there is an easier proof. My proof is a little ...
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Exterior sphere condition and Lipschitz Condition on the boundary for elliptic PDE

I am currently stuck on a problem. Here is the description: Assume $\Omega$ is a $C^1$, bounded domain that satisfies the exterior sphere condition: For every point $x_0 \in \partial \Omega$, there ...
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Show this function is Lipschitz

Consider the following function $$f(x)= \begin{cases} e^{x/2} \qquad x \leq 0 \\ x + e^{-x} \qquad x >0 \end{cases}$$ I need to show it's Lipschitz with constant $1$: $|T(x)-T(y)|<|x-y|$. For ...
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Approximation of Lipschitz function by smooth functions

I'm reading the book Measure theory and fine properties of functions by Evans and Gariepy, but there is a step in the Sobolev extension theorem (Theorem 4.7) that I don't understand. Concretely, the ...
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Can i represent derivative as Lipschitz constant, if a function defined on interval (a,b]?

I am working with some piecewise linear functions and its' intervals defined as $I_j$=($x_j$,$x_{j+1}$]. I want to use the derivatives of sub-functions as Lipschitz constants, but in order to do that, ...
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Supremum of uniformly Lipschitz functions is Lipschitz

If we have a set of functions $f_i$ such that every $f_i$ has a common Lipschitz constant $M$, is it true that the supremum of these functions at some $x$ is also Lipschitz? Assume that we can ...
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Locally and globally Lipschitz

I have to see if the function $f(t,x)=x^n, \hspace{2mm}n>1$ and $x\in \mathbb{R}$ satisfies the condition of globally or locally Lipschitz with respect to the second variable $x$. I understand the ...
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Locally Lipschitz operator in $H^1(0,1) \times H^1(0,1).$

I have been trying to show that a map $T$ defined as $$T(f,g) = \dfrac{f|f|}{g}, \qquad f,g \in H^1((0,1))$$ is locally Lipschitz. I am not very sure how to do it since I am not even sure that $T$ is ...
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Lebesgue differentiation theorem for monotone functions via Vitali covering lemma

I was reading LECTURES ON LIPSCHITZ ANALYSIS by Juha Heinonen and at the Theorem 3.2 he gives a proof of Lebesgue Differentiation Theorem for monotone functions. He says that we can easily (using the ...
The solution is found here: sum and product of Lipschitz functions I'm not sure from where the following line $$|f(x)g(x) - f(x)g(y)| \leq M|g(x) - g(y)|$$ comes from. I tried looking at assumption ...