For which values of $c$ is the function $f(x)$ in $W^{1,p}_{mathrm{loc}} (mathbb{R})$?












2












$begingroup$


We're given the function



$$f(x) = begin{cases} 2 sin(x) + 3, & x> 0 \
-2 sin(x) + c, & x leq 0
end{cases} \$$



and asked to find the values of $c$ for which $f in W^{1,p}_{mathrm{loc}}$.



To start, I've found the distributional derivative of $f$ to be



$$f'(x) = begin{cases} 2 cos(x), & x> 0 \
-2 cos(x), & x leq 0
end{cases} \$$



and I want to find values of $c$ for which $leftlVert frightrVert_{L^p(K)} < infty$ where $K$ is any closed ball in $mathbb{R}$. But I'm stuck on that computation.










share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    We're given the function



    $$f(x) = begin{cases} 2 sin(x) + 3, & x> 0 \
    -2 sin(x) + c, & x leq 0
    end{cases} \$$



    and asked to find the values of $c$ for which $f in W^{1,p}_{mathrm{loc}}$.



    To start, I've found the distributional derivative of $f$ to be



    $$f'(x) = begin{cases} 2 cos(x), & x> 0 \
    -2 cos(x), & x leq 0
    end{cases} \$$



    and I want to find values of $c$ for which $leftlVert frightrVert_{L^p(K)} < infty$ where $K$ is any closed ball in $mathbb{R}$. But I'm stuck on that computation.










    share|cite|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      We're given the function



      $$f(x) = begin{cases} 2 sin(x) + 3, & x> 0 \
      -2 sin(x) + c, & x leq 0
      end{cases} \$$



      and asked to find the values of $c$ for which $f in W^{1,p}_{mathrm{loc}}$.



      To start, I've found the distributional derivative of $f$ to be



      $$f'(x) = begin{cases} 2 cos(x), & x> 0 \
      -2 cos(x), & x leq 0
      end{cases} \$$



      and I want to find values of $c$ for which $leftlVert frightrVert_{L^p(K)} < infty$ where $K$ is any closed ball in $mathbb{R}$. But I'm stuck on that computation.










      share|cite|improve this question











      $endgroup$




      We're given the function



      $$f(x) = begin{cases} 2 sin(x) + 3, & x> 0 \
      -2 sin(x) + c, & x leq 0
      end{cases} \$$



      and asked to find the values of $c$ for which $f in W^{1,p}_{mathrm{loc}}$.



      To start, I've found the distributional derivative of $f$ to be



      $$f'(x) = begin{cases} 2 cos(x), & x> 0 \
      -2 cos(x), & x leq 0
      end{cases} \$$



      and I want to find values of $c$ for which $leftlVert frightrVert_{L^p(K)} < infty$ where $K$ is any closed ball in $mathbb{R}$. But I'm stuck on that computation.







      functional-analysis sobolev-spaces lp-spaces






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 2 at 17:25









      Davide Giraudo

      128k17154268




      128k17154268










      asked Jan 2 at 3:45









      kkckkc

      17011




      17011






















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          $begingroup$

          Your claimed distributional derivative is not correct in general. Indeed you may pick up a extra Dirac delta term arising from the fact that,
          $$frac{mathrm{d}}{mathrm{d}t}, 1_{(-infty,0]} = delta_0.$$



          The following rules out all but possibly one value of $c.$




          Claim: If $f in W^{1,p}_{mathrm{loc}}(mathbb R) subset W^{1,1}_{mathrm{loc}}(mathbb R),$ then $f$ is continuous (more precisely it admits a continuous representative).




          Proof of claim: Note that if $g in C^{infty}_c(mathbb R),$ then by the fundamental theorem of calculus and $a < x < b,$
          begin{align*}
          |g(x)| &= left|g(a)+int_{a}^x g'(t) ,mathrm{d}t right| \
          &leq frac1{b-a}lVert g rVert_{L^1([a,b])} + frac1{b-a}int_a^b |g(a)-g(x)|,mathrm{d}t + lVert g' rVert_{L^1((a,b))} \
          &leq C_{a,b} lVert g rVert_{W^{1,1}([a,b])}.
          end{align*}

          So $lVert g rVert_{L^{infty}([a,b])} leq C lVert g rVert_{W^{1,1}([a,b])}$ for all $g in C^{infty}([a,b])$ and $[a,b] subset mathbb R.$ By approximating $f$ in $W^{1,1}(mathbb R)$ via mollification, we get a sequence of smooth functions which converges locally uniformly to $f$ by the above estimate. Hence $f$ is continuous in $mathbb R.$ $square$



          Now your $f$ as defined is continuous if and only if $c=3.$ So we get $f not in W^{1,p}_{mathrm{loc}}(mathbb R)$ if $c neq 3.$





          Now what happens if $c=3$? We first need to show that $f'$ is what you claim, so let
          $$g(x) = begin{cases} 2 cos(x) &text{if } x> 0 \
          -2 cos(x) &text{if } x leq 0
          end{cases} \$$

          be the candidate for $f'$ and fix an arbitrary $varphi in C^{infty}_c(mathbb R).$ Then as $f$ is continuous on $mathbb R$ and $C^1$ with derivative $g$ on $mathbb R setminus {0}$, integrating by parts on $(-infty,0)$ and $(0,infty)$ we get,
          begin{align*}
          int_{-infty}^{infty} varphi'(t) f(t) ,mathrm{d}t &= int_{-infty}^0 varphi'(t) f(t) + int_0^{infty} varphi'(t) f(t),mathrm{d}t \
          &= left(varphi'(0)f(0) - int_{-infty}^0 varphi(t)g(t) ,mathrm{d}tright) + left(-varphi'(0)f(0) - int_0^{infty} varphi(t)g(t) ,mathrm{d}tright) \
          &= - int_{-infty}^{infty} varphi(t)g(t).
          end{align*}

          Thus $f$ is weakly differentiable with weak derivative $f' = g.$ Note that we needed continuity of $f$ at $0$ to integrate by parts.



          Finally as both $f$ and $f'$ are bounded, we get $f, f' in L^{infty}(mathbb R)$ and hence $f,f' in L^p_{mathrm{loc}}(mathbb R)$ for all $p$ by Hölder. Hence $f in W^{1,p}_{mathrm{loc}}(mathbb R)$ for all $1 leq p leq infty$ when $c=3.$






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            $begingroup$

            Your claimed distributional derivative is not correct in general. Indeed you may pick up a extra Dirac delta term arising from the fact that,
            $$frac{mathrm{d}}{mathrm{d}t}, 1_{(-infty,0]} = delta_0.$$



            The following rules out all but possibly one value of $c.$




            Claim: If $f in W^{1,p}_{mathrm{loc}}(mathbb R) subset W^{1,1}_{mathrm{loc}}(mathbb R),$ then $f$ is continuous (more precisely it admits a continuous representative).




            Proof of claim: Note that if $g in C^{infty}_c(mathbb R),$ then by the fundamental theorem of calculus and $a < x < b,$
            begin{align*}
            |g(x)| &= left|g(a)+int_{a}^x g'(t) ,mathrm{d}t right| \
            &leq frac1{b-a}lVert g rVert_{L^1([a,b])} + frac1{b-a}int_a^b |g(a)-g(x)|,mathrm{d}t + lVert g' rVert_{L^1((a,b))} \
            &leq C_{a,b} lVert g rVert_{W^{1,1}([a,b])}.
            end{align*}

            So $lVert g rVert_{L^{infty}([a,b])} leq C lVert g rVert_{W^{1,1}([a,b])}$ for all $g in C^{infty}([a,b])$ and $[a,b] subset mathbb R.$ By approximating $f$ in $W^{1,1}(mathbb R)$ via mollification, we get a sequence of smooth functions which converges locally uniformly to $f$ by the above estimate. Hence $f$ is continuous in $mathbb R.$ $square$



            Now your $f$ as defined is continuous if and only if $c=3.$ So we get $f not in W^{1,p}_{mathrm{loc}}(mathbb R)$ if $c neq 3.$





            Now what happens if $c=3$? We first need to show that $f'$ is what you claim, so let
            $$g(x) = begin{cases} 2 cos(x) &text{if } x> 0 \
            -2 cos(x) &text{if } x leq 0
            end{cases} \$$

            be the candidate for $f'$ and fix an arbitrary $varphi in C^{infty}_c(mathbb R).$ Then as $f$ is continuous on $mathbb R$ and $C^1$ with derivative $g$ on $mathbb R setminus {0}$, integrating by parts on $(-infty,0)$ and $(0,infty)$ we get,
            begin{align*}
            int_{-infty}^{infty} varphi'(t) f(t) ,mathrm{d}t &= int_{-infty}^0 varphi'(t) f(t) + int_0^{infty} varphi'(t) f(t),mathrm{d}t \
            &= left(varphi'(0)f(0) - int_{-infty}^0 varphi(t)g(t) ,mathrm{d}tright) + left(-varphi'(0)f(0) - int_0^{infty} varphi(t)g(t) ,mathrm{d}tright) \
            &= - int_{-infty}^{infty} varphi(t)g(t).
            end{align*}

            Thus $f$ is weakly differentiable with weak derivative $f' = g.$ Note that we needed continuity of $f$ at $0$ to integrate by parts.



            Finally as both $f$ and $f'$ are bounded, we get $f, f' in L^{infty}(mathbb R)$ and hence $f,f' in L^p_{mathrm{loc}}(mathbb R)$ for all $p$ by Hölder. Hence $f in W^{1,p}_{mathrm{loc}}(mathbb R)$ for all $1 leq p leq infty$ when $c=3.$






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              Your claimed distributional derivative is not correct in general. Indeed you may pick up a extra Dirac delta term arising from the fact that,
              $$frac{mathrm{d}}{mathrm{d}t}, 1_{(-infty,0]} = delta_0.$$



              The following rules out all but possibly one value of $c.$




              Claim: If $f in W^{1,p}_{mathrm{loc}}(mathbb R) subset W^{1,1}_{mathrm{loc}}(mathbb R),$ then $f$ is continuous (more precisely it admits a continuous representative).




              Proof of claim: Note that if $g in C^{infty}_c(mathbb R),$ then by the fundamental theorem of calculus and $a < x < b,$
              begin{align*}
              |g(x)| &= left|g(a)+int_{a}^x g'(t) ,mathrm{d}t right| \
              &leq frac1{b-a}lVert g rVert_{L^1([a,b])} + frac1{b-a}int_a^b |g(a)-g(x)|,mathrm{d}t + lVert g' rVert_{L^1((a,b))} \
              &leq C_{a,b} lVert g rVert_{W^{1,1}([a,b])}.
              end{align*}

              So $lVert g rVert_{L^{infty}([a,b])} leq C lVert g rVert_{W^{1,1}([a,b])}$ for all $g in C^{infty}([a,b])$ and $[a,b] subset mathbb R.$ By approximating $f$ in $W^{1,1}(mathbb R)$ via mollification, we get a sequence of smooth functions which converges locally uniformly to $f$ by the above estimate. Hence $f$ is continuous in $mathbb R.$ $square$



              Now your $f$ as defined is continuous if and only if $c=3.$ So we get $f not in W^{1,p}_{mathrm{loc}}(mathbb R)$ if $c neq 3.$





              Now what happens if $c=3$? We first need to show that $f'$ is what you claim, so let
              $$g(x) = begin{cases} 2 cos(x) &text{if } x> 0 \
              -2 cos(x) &text{if } x leq 0
              end{cases} \$$

              be the candidate for $f'$ and fix an arbitrary $varphi in C^{infty}_c(mathbb R).$ Then as $f$ is continuous on $mathbb R$ and $C^1$ with derivative $g$ on $mathbb R setminus {0}$, integrating by parts on $(-infty,0)$ and $(0,infty)$ we get,
              begin{align*}
              int_{-infty}^{infty} varphi'(t) f(t) ,mathrm{d}t &= int_{-infty}^0 varphi'(t) f(t) + int_0^{infty} varphi'(t) f(t),mathrm{d}t \
              &= left(varphi'(0)f(0) - int_{-infty}^0 varphi(t)g(t) ,mathrm{d}tright) + left(-varphi'(0)f(0) - int_0^{infty} varphi(t)g(t) ,mathrm{d}tright) \
              &= - int_{-infty}^{infty} varphi(t)g(t).
              end{align*}

              Thus $f$ is weakly differentiable with weak derivative $f' = g.$ Note that we needed continuity of $f$ at $0$ to integrate by parts.



              Finally as both $f$ and $f'$ are bounded, we get $f, f' in L^{infty}(mathbb R)$ and hence $f,f' in L^p_{mathrm{loc}}(mathbb R)$ for all $p$ by Hölder. Hence $f in W^{1,p}_{mathrm{loc}}(mathbb R)$ for all $1 leq p leq infty$ when $c=3.$






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                Your claimed distributional derivative is not correct in general. Indeed you may pick up a extra Dirac delta term arising from the fact that,
                $$frac{mathrm{d}}{mathrm{d}t}, 1_{(-infty,0]} = delta_0.$$



                The following rules out all but possibly one value of $c.$




                Claim: If $f in W^{1,p}_{mathrm{loc}}(mathbb R) subset W^{1,1}_{mathrm{loc}}(mathbb R),$ then $f$ is continuous (more precisely it admits a continuous representative).




                Proof of claim: Note that if $g in C^{infty}_c(mathbb R),$ then by the fundamental theorem of calculus and $a < x < b,$
                begin{align*}
                |g(x)| &= left|g(a)+int_{a}^x g'(t) ,mathrm{d}t right| \
                &leq frac1{b-a}lVert g rVert_{L^1([a,b])} + frac1{b-a}int_a^b |g(a)-g(x)|,mathrm{d}t + lVert g' rVert_{L^1((a,b))} \
                &leq C_{a,b} lVert g rVert_{W^{1,1}([a,b])}.
                end{align*}

                So $lVert g rVert_{L^{infty}([a,b])} leq C lVert g rVert_{W^{1,1}([a,b])}$ for all $g in C^{infty}([a,b])$ and $[a,b] subset mathbb R.$ By approximating $f$ in $W^{1,1}(mathbb R)$ via mollification, we get a sequence of smooth functions which converges locally uniformly to $f$ by the above estimate. Hence $f$ is continuous in $mathbb R.$ $square$



                Now your $f$ as defined is continuous if and only if $c=3.$ So we get $f not in W^{1,p}_{mathrm{loc}}(mathbb R)$ if $c neq 3.$





                Now what happens if $c=3$? We first need to show that $f'$ is what you claim, so let
                $$g(x) = begin{cases} 2 cos(x) &text{if } x> 0 \
                -2 cos(x) &text{if } x leq 0
                end{cases} \$$

                be the candidate for $f'$ and fix an arbitrary $varphi in C^{infty}_c(mathbb R).$ Then as $f$ is continuous on $mathbb R$ and $C^1$ with derivative $g$ on $mathbb R setminus {0}$, integrating by parts on $(-infty,0)$ and $(0,infty)$ we get,
                begin{align*}
                int_{-infty}^{infty} varphi'(t) f(t) ,mathrm{d}t &= int_{-infty}^0 varphi'(t) f(t) + int_0^{infty} varphi'(t) f(t),mathrm{d}t \
                &= left(varphi'(0)f(0) - int_{-infty}^0 varphi(t)g(t) ,mathrm{d}tright) + left(-varphi'(0)f(0) - int_0^{infty} varphi(t)g(t) ,mathrm{d}tright) \
                &= - int_{-infty}^{infty} varphi(t)g(t).
                end{align*}

                Thus $f$ is weakly differentiable with weak derivative $f' = g.$ Note that we needed continuity of $f$ at $0$ to integrate by parts.



                Finally as both $f$ and $f'$ are bounded, we get $f, f' in L^{infty}(mathbb R)$ and hence $f,f' in L^p_{mathrm{loc}}(mathbb R)$ for all $p$ by Hölder. Hence $f in W^{1,p}_{mathrm{loc}}(mathbb R)$ for all $1 leq p leq infty$ when $c=3.$






                share|cite|improve this answer









                $endgroup$



                Your claimed distributional derivative is not correct in general. Indeed you may pick up a extra Dirac delta term arising from the fact that,
                $$frac{mathrm{d}}{mathrm{d}t}, 1_{(-infty,0]} = delta_0.$$



                The following rules out all but possibly one value of $c.$




                Claim: If $f in W^{1,p}_{mathrm{loc}}(mathbb R) subset W^{1,1}_{mathrm{loc}}(mathbb R),$ then $f$ is continuous (more precisely it admits a continuous representative).




                Proof of claim: Note that if $g in C^{infty}_c(mathbb R),$ then by the fundamental theorem of calculus and $a < x < b,$
                begin{align*}
                |g(x)| &= left|g(a)+int_{a}^x g'(t) ,mathrm{d}t right| \
                &leq frac1{b-a}lVert g rVert_{L^1([a,b])} + frac1{b-a}int_a^b |g(a)-g(x)|,mathrm{d}t + lVert g' rVert_{L^1((a,b))} \
                &leq C_{a,b} lVert g rVert_{W^{1,1}([a,b])}.
                end{align*}

                So $lVert g rVert_{L^{infty}([a,b])} leq C lVert g rVert_{W^{1,1}([a,b])}$ for all $g in C^{infty}([a,b])$ and $[a,b] subset mathbb R.$ By approximating $f$ in $W^{1,1}(mathbb R)$ via mollification, we get a sequence of smooth functions which converges locally uniformly to $f$ by the above estimate. Hence $f$ is continuous in $mathbb R.$ $square$



                Now your $f$ as defined is continuous if and only if $c=3.$ So we get $f not in W^{1,p}_{mathrm{loc}}(mathbb R)$ if $c neq 3.$





                Now what happens if $c=3$? We first need to show that $f'$ is what you claim, so let
                $$g(x) = begin{cases} 2 cos(x) &text{if } x> 0 \
                -2 cos(x) &text{if } x leq 0
                end{cases} \$$

                be the candidate for $f'$ and fix an arbitrary $varphi in C^{infty}_c(mathbb R).$ Then as $f$ is continuous on $mathbb R$ and $C^1$ with derivative $g$ on $mathbb R setminus {0}$, integrating by parts on $(-infty,0)$ and $(0,infty)$ we get,
                begin{align*}
                int_{-infty}^{infty} varphi'(t) f(t) ,mathrm{d}t &= int_{-infty}^0 varphi'(t) f(t) + int_0^{infty} varphi'(t) f(t),mathrm{d}t \
                &= left(varphi'(0)f(0) - int_{-infty}^0 varphi(t)g(t) ,mathrm{d}tright) + left(-varphi'(0)f(0) - int_0^{infty} varphi(t)g(t) ,mathrm{d}tright) \
                &= - int_{-infty}^{infty} varphi(t)g(t).
                end{align*}

                Thus $f$ is weakly differentiable with weak derivative $f' = g.$ Note that we needed continuity of $f$ at $0$ to integrate by parts.



                Finally as both $f$ and $f'$ are bounded, we get $f, f' in L^{infty}(mathbb R)$ and hence $f,f' in L^p_{mathrm{loc}}(mathbb R)$ for all $p$ by Hölder. Hence $f in W^{1,p}_{mathrm{loc}}(mathbb R)$ for all $1 leq p leq infty$ when $c=3.$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 4 at 15:00









                ktoiktoi

                2,4361618




                2,4361618






























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