Let's say we are on some bounded domain $K\subset\mathbb{R}^n$. Under which conditions is $C^1(K)$ dense in $W^{1,p}(K)$? I would like to approximate a function in $W^{1,p}(K)$ by bounded $C^1(K)$ functions but am not sure whether or not the boundedness of the $C^1(K)$ function suffices in order to do so! I know that this often (for unbounded functions) is only possible under certain assumptions on the domain such as a Lipschitz boundary etc. - but this is not given here. Thanks in advance!
2026-03-25 22:23:58.1774477438
Approximation of bounded, differentiable function in $W^{1,p}$
140 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in SOBOLEV-SPACES
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- $\mbox{Cap}_p$-measurability
- If $u\in W^{1,p}(\Omega )$ is s.t. $\nabla u=0$ then $u$ is constant a.e.
- Weak formulation of Robin boundary condition problem
- Variational Formulation - inhomogeneous Neumann boundary
- Why the Sobolev space $W^{1,2}(M,N)$ weak-sequencially closed in $W^{1,2}(\mathbb R^K)$?
- Sobolev space $H^s(Q)$ is Hilbert
- Duhamel's principle for heat equation.
- How to define discrete Sobolev dual norm so that it can be computed?
- Weakly sequentially continuous maps
Related Questions in APPROXIMATION
- Does approximation usually exclude equality?
- Approximate spline equation with Wolfram Mathematica
- Solving Equation with Euler's Number
- Approximate derivative in midpoint rule error with notation of Big O
- An inequality involving $\int_0^{\frac{\pi}{2}}\sqrt{\sin x}\:dx $
- On the rate of convergence of the central limit theorem
- Is there any exponential function that can approximate $\frac{1}{x}$?
- Gamma distribution to normal approximation
- Product and Quotient Rule proof using linearisation
- Best approximation of a function out of a closed subset
Related Questions in DENSE-SUBSPACES
- Understanding dense sets
- Is it possible to define a topology on the real line such that 0 and non-zero integers are dense but no finite subset of non-zero integers is dense?
- Characterization of density and adherence in topological space.
- Extension of non-first countable space
- How can I show that this set is not dense in $S^1\times S^1$?
- Rudin's RCA Theorem $4.18$.
- Hausdorff measure of uncountable dense subsets
- If $X$ is locally compact, then $\mathrm{Lib}_c (X)$ is dense in $L^p (X, \mu)$ for $p \in [1, \infty)$
- Proof on why solution to a linear system Ax=b belong to the set of initial abitrary guess plus the Krylov subspace of the matrix & intitial residual?
- Nowhere dense subsets of a dense subspace are nowhere dense in the whole space and vice versa
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
This is the Meyer-Serrin theorem from their famous $H=W$ paper: The set $C^\infty(K) \cap W^{m,p}(K)$ is dense in $W^{m,p}(K)$. No assumptions on $K$ needed. However, note that $C^\infty(K)$ functions might behave very badly close to the boundary, i.e., $u(x) = 1/x$ is in $C^\infty(0,1)$.
https://en.wikipedia.org/wiki/Meyers%E2%80%93Serrin_theorem